Math Problems Vault

Problem 381

The function $f(x)$ is defined by the following conditions:

If $x=\dfrac {m}{n}$ , then $f(x)=n$ , where $m$ and $n$ are coprime integers with $n>0$ ;

If $x$ is irrational, then $f(x)=0$ .

Prove that this function is finite at every point $x$ , but is not bounded (that is, it is unbounded in any neighborhood of that point).

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Problem 382

(a)(b)

If the function $f(x)$ is defined and bounded at every point of: (a) an open interval, (б) a closed interval, is this function necessarily bounded on the given interval, or on the corresponding closed interval?

Give suitable examples.

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(a)

(b)

Problem 383

Prove that the function $f(x)=\dfrac {1+x^{2}}{1+x^{4}}$ is bounded on the interval $-\infty <x<+\infty$ .

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Problem 384

Prove that the function $f(x)=\dfrac {1}{x}\cos \dfrac {1}{x}$ is unbounded in any neighborhood of the point $x=0$ , but does not become infinite as $x \to 0$ .

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Problem 385

Investigate the boundedness of the function $f(x)=\ln x \cdot \sin^{2}\dfrac {\pi }{x}$ on the interval $0<x<\varepsilon$ .

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Problem 386

Prove that the function $f(x)=\dfrac {x}{1+x}$ has infimum $m_{0}=0$ and supremum $M_{0}=1$ on the domain $0 \leqslant x<+\infty$ .

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Problem 387

The function $f(x)$ is defined on the closed interval $[a,b]$ and increases monotonically. What are the infimum and supremum of the function on this closed interval equal to?

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Problem 388

Find the infimum and supremum of the function: $f(x)=x^{2}$ on $(-2,5)$ .

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Problem 389

Find the infimum and supremum of the function: $f(x)=\dfrac {1}{1+x^{2}}$ on $(-\infty ,+\infty )$ .

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Problem 390

Find the infimum and supremum of the function: $f(x)=\dfrac {2x}{1+x^{2}}$ on $(0,+\infty )$ .

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Problem 391

Find the infimum and supremum of the function: $f(x)=x+\dfrac {1}{x}$ on $(0,+\infty )$ .

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Problem 392

Find the infimum and supremum of the function: $f(x)=\sin x$ on $(0,+\infty )$ .

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Problem 393

Find the infimum and supremum of the function: $f(x)=\sin x+\cos x$ on $[0,2\pi ]$ .

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Problem 394

Find the infimum and supremum of the function: $f(x)=2^{x}$ on $(-1,2)$ .

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Problem 395

(a)(b)

Find the infimum and supremum of the function: $f(x)=[x]$ ; (a) on $(0,2)$ , (б) on $[0,2]$ .

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(a)

(b)

Problem 396

Find the infimum and supremum of the function: $f(x)=x-[x]$ on $[0,1]$ .

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Problem 397

(a)(b)(c)(d)

Find the amplitude (oscillation) of the function $f(x)=x^{2}$ on the following intervals:

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(a)

$(1,3)$ ;

(b)

$(1.9,2.1)$ ;

(c)

$(1.99,2.01)$ ;

(d)

$(1.999,2.001)$ .

Problem 398

(a)(b)(c)(d)

Find the amplitude (oscillation) of the function

f(x)=\operatorname {arctg}\dfrac {1}{x}

on the following intervals:

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(a)

$(-1,+1)$ ;

(b)

$(-0.1,0.1)$ ;

(c)

$(-0.01,0.01)$ ;

(d)

$(-0.001,0.001)$ .

Problem 399

(a)(b)

Let $m[f]$ and $M[f]$ denote respectively the infimum and supremum of the function $f(x)$ on the interval $(a,b)$ .

Prove that if $f_{1}(x)$ and $f_{2}(x)$ are functions defined on $(a,b)$ , then

m[f_{1}+f_{2}] \geqslant m[f_{1}]+m[f_{2}]

and

M[f_{1}+f_{2}] \leqslant M[f_{1}]+M[f_{2}] .

Give examples of functions $f_{1}(x)$ and $f_{2}(x)$ such that in the last two relations: (a) equality holds, (б) strict inequality holds.

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(a)

(b)

Problem 400

(a)(b)

Let the function $f(x)$ be defined on the domain $[a,+\infty )$ , and bounded on every closed interval $[a,b]$ . Suppose

m(x)=\inf _{a \leqslant \xi \leqslant x}f(\xi )

and

M(x)=\sup _{a \leqslant \xi \leqslant x}f(\xi ) .

Draw the graphs of $y=m(x)$ and $y=M(x)$ , given

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(a)

$f(x)=\sin x$ ;

(b)

$f(x)=\cos x$ .

Problem 401

Using the $\varepsilon -\delta$ argument, prove that

\lim _{x\to 2}x^{2}=4 .

Fill in the following table:

$\varepsilon$	$0.1$	$0.01$	$0.001$	$0.0001$	$\cdots$
$\delta$					$\cdots$

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Problem 402

Using the $E-\delta$ argument, prove that

\lim _{x\to 1}\dfrac {1}{(1-x)^{2}}=+\infty .

Fill in the following table:

$E$	$10$	$100$	$1000$	$10000$	$\cdots$
$\delta$					$\cdots$

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Problem 403

(a)(b)(c)

Express the following statements using inequalities:

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(a)

$\lim \limits_{x\to a}f(x)=b$ ;

(b)

$\lim \limits_{x\to a-0}f(x)=b$ ;

(c)

$\lim \limits_{x\to a+0}f(x)=b$ .

Give suitable examples.

Problem 404

(a)(b)(c)

Express the following statements using inequalities, and give suitable examples:

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(a)

$\lim \limits_{x\to \infty }f(x)=b$ ;

(b)

$\lim \limits_{x\to -\infty }f(x)=b$ ;

(c)

$\lim \limits_{x\to +\infty }f(x)=b$ .

Problem 405

Express the following statements using inequalities, and give suitable examples:

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(a)

$\lim \limits_{x\to a}f(x)=\infty$ ;

(b)

$\lim \limits_{x\to a}f(x)=-\infty$ ;

(c)

$\lim \limits_{x\to a}f(x)=+\infty$ ;

(d)

$\lim \limits_{x\to a-0}f(x)=\infty$ ;

(e)

$\lim \limits_{x\to a-0}f(x)=-\infty$ ;

(f)

$\lim \limits_{x\to a-0}f(x)=+\infty$ ;

(g)

$\lim \limits_{x\to a+0}f(x)=\infty$ ;

(h)

$\lim \limits_{x\to a+0}f(x)=-\infty$ ;

(i)

$\lim \limits_{x\to a+0}f(x)=+\infty$ .

Problem 406

Express the following statements using inequalities, and give suitable examples:

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(a)

$\lim \limits_{x\to \infty }f(x)=\infty$ ;

(b)

$\lim \limits_{x\to \infty }f(x)=-\infty$ ;

(c)

$\lim \limits_{x\to \infty }f(x)=+\infty$ ;

(d)

$\lim \limits_{x\to -\infty }f(x)=\infty$ ;

(e)

$\lim \limits_{x\to -\infty }f(x)=-\infty$ ;

(f)

$\lim \limits_{x\to -\infty }f(x)=+\infty$ ;

(g)

$\lim \limits_{x\to +\infty }f(x)=\infty$ ;

(h)

$\lim \limits_{x\to +\infty }f(x)=-\infty$ ;

(i)

$\lim \limits_{x\to +\infty }f(x)=+\infty$ .

Problem 407

Let $y=f(x)$ . Express the following situations using inequalities, and give suitable examples:

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(a)

as $x\to a$ , $y\to b-0$ ;

(b)

as $x\to a-0$ , $y\to b-0$ ;

(c)

as $x\to a+0$ , $y\to b-0$ ;

(d)

as $x\to a$ , $y\to b+0$ ;

(e)

as $x\to a-0$ , $y\to b+0$ ;

(f)

as $x\to a+0$ , $y\to b+0$ ;

(g)

as $x\to \infty$ , $y\to b-0$ ;

(h)

as $x\to -\infty$ , $y\to b-0$ ;

(i)

as $x\to +\infty$ , $y\to b-0$ ;

(j)

as $x\to +\infty$ , $y\to b+0$ ;

(k)

as $x\to -\infty$ , $y\to b+0$ ;

(l)

as $x\to +\infty$ , $y\to b+0$ .

Problem 408

Let

p(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n},

where $a_{i}(i=0,1,\cdots ,n)$ are real numbers. Prove that $\lim \limits_{x\to \infty }\left|p(x)\right|=+\infty$ .

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Problem 409

Let

R(x)=\frac{a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}}{b_{0}x^{m}+b_{1}x^{m-1}+\cdots +b_{m}} ,

where $a_{0}\neq 0,b_{0}\neq 0$ . Prove that:

\lim _{x\to \infty }R(x)= \begin{cases} \infty , & \text{if } n>m ; \\ \dfrac {a_{0}}{b_{0}}, & \text{if } n=m ; \\ 0, & \text{if } n<m . \end{cases}

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Problem 410

Let

R(x)=\frac{P(x)}{Q(x)} ,

where $P(x)$ and $Q(x)$ are polynomials in $x$ , and $P(a)=Q(a)=0$ . What possible values can

\lim _{x\to a}\frac{P(x)}{Q(x)}?

have?

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Problem 411

(a)(b)(c)

Find the values of the following expressions:

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(a)

$\lim \limits_{x\to 0}\dfrac {x^{2}-1}{2x^{2}-x-1}$ ;

(b)

$\lim \limits_{x\to 1}\dfrac {x^{2}-1}{2x^{2}-x-1}$ ;

(c)

$\lim \limits_{x\to \infty }\dfrac {x^{2}-1}{2x^{2}-x-1}$ .

Problem 412

Find $\lim \limits_{x\to 0}\dfrac {(1+x)(1+2x)(1+3x)-1}{x}$ .

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Problem 413

Find $\lim \limits_{x\to 0}\dfrac {(1+x)^{5}-(1+5x)}{x^{2}+x^{5}}$ .

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Problem 414

Find $\lim \limits_{x\to 0}\dfrac {(1+mx)^{n}-(1+nx)^{m}}{x^{2}}$ ( $m$ and $n$ are natural numbers).

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Problem 415

Find $\lim \limits_{x\to \infty }\dfrac {(x-1)(x-2)(x-3)(x-4)(x-5)}{(5x-1)^{5}}$ .

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Problem 416

Find $\lim \limits_{x\to \infty }\dfrac {(2x-3)^{20}(3x+2)^{30}}{(2x+1)^{50}}$ .

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Problem 417

Find $\lim \limits_{x\to \infty }\dfrac {(x+1)(x^{2}+1)\cdots (x^{n}+1)}{\left[(nx)^{n}+1\right]^{\frac{n+1}{2}}}$ .

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Problem 418

Find $\lim \limits_{x\to 3}\dfrac {x^{2}-5x+6}{x^{2}-8x+15}$ .

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Problem 419

Find $\lim \limits_{x\to 1}\dfrac {x^{3}-3x+2}{x^{4}-4x+3}$ .

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Problem 420

Find $\lim \limits_{x\to 1}\dfrac {x^{3}-3x+2}{x^{4}-x^{3}-x+1}$ .

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Problem 421

Find $\lim \limits_{x\to 2}\dfrac {x^{3}-2x^{2}-4x+8}{x^{4}-8x^{2}+16}$ .

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Problem 422

Find $\lim \limits_{x\to -1}\dfrac {x^{3}-2x-1}{x^{5}-2x-1}$ .

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Problem 423

Find $\lim \limits_{x\to 2}\dfrac {(x^{2}-x-2)^{20}}{(x^{3}-12x+16)^{10}}$ .

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Problem 424

Find $\lim \limits_{x\to 1}\dfrac {x+x^{2}+\cdots +x^{n}-n}{x-1}$ .

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Problem 424.1

[NO TASK]

Problem 425

Find $\lim \limits_{x\to 1}\dfrac {x^{m}-1}{x^{n}-1}$ ( $m$ and $n$ are natural numbers).

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Problem 426

Find $\lim \limits_{x\to a}\dfrac {(x^{n}-a^{n})-na^{n-1}(x-a)}{(x-a)^{2}}$ ( $n$ denotes a natural number).

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Problem 427

Find $\lim \limits_{x\to 1}\dfrac {x^{n+1}-(n+1)x+n}{(x-1)^{2}}$ ( $n$ denotes a natural number).

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Problem 428

Find $\lim \limits_{x\to 1}\left(\dfrac {m}{1-x^{m}}-\dfrac {n}{1-x^{n}}\right)$ ( $m$ and $n$ are natural numbers).

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Problem 429

Find $\lim \limits_{n\to \infty }\dfrac {1}{n}\left[\left(x+\dfrac {a}{n}\right)+\left(x+\dfrac {2a}{n}\right)+\cdots +\left(x+\dfrac {(n-1)a}{n}\right)\right]$ .

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Problem 430

Find $\lim \limits_{n\to \infty }\dfrac {1}{n}\left[\left(x+\dfrac {a}{n}\right)^{2}+\left(x+\dfrac {2a}{n}\right)^{2}+\cdots +\left(x+\dfrac {(n-1)a}{n}\right)^{2}\right]$ .

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Problem 431

Find $\lim \limits_{n\to \infty }\dfrac {1^{2}+3^{2}+\cdots +(2n-1)^{2}}{2^{2}+4^{2}+\cdots +(2n)^{2}}$ .

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Problem 432

Find $\lim \limits_{n\to \infty }\left(\dfrac {1^{3}+2^{3}+\cdots +n^{3}}{n^{3}}-\dfrac {n}{4}\right)$ .

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Problem 433

Find $\lim \limits_{n\to \infty }\dfrac {1^{3}+4^{3}+7^{3}+\cdots +(3n-2)^{3}}{\left[1+4+7+\cdots +(3n-2)\right]^{2}}$ .

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Problem 434

Take the area of the curvilinear triangle $OAM$ (Figure 1.237) bounded by the parabola $y=b\left(\dfrac {x}{a}\right)^{2}$ , the $Ox$ axis, and the line $x=a$ , to be the limit as $n\to \infty$ of the sum of the areas of inscribed rectangles each with base $\dfrac {a}{n}$ ; find this area.

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Problem 435

Find the limit: $\lim \limits_{x\to +\infty }\dfrac {\sqrt{x+\sqrt{x+\sqrt{x}}}}{\sqrt{x+1}}$ .

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Problem 436

Find the limit: $\lim \limits_{x\to +\infty }\dfrac {\sqrt{x}+\sqrt[3]{x}+\sqrt[4]{x}}{\sqrt{2x+1}}$ .

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Problem 437

Find the limit: $\lim \limits_{x\to 4}\dfrac {\sqrt{1+2x}-3}{\sqrt{x}-2}$ .

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Problem 438

Find the limit: $\lim \limits_{x\to -8}\dfrac {\sqrt{1-x}-3}{2+\sqrt[3]{x}}$ .

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Problem 439

Find the limit: $\lim \limits_{x\to a}\dfrac {\sqrt{x}-\sqrt{a}+\sqrt{x-a}}{\sqrt{x^{2}-a^{2}}}$ ( $a>0$ ).

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Problem 440

Find the limit: $\lim \limits_{x\to 3}\dfrac {\sqrt{x+13}-2\sqrt{x+1}}{x^{2}-9}$ .

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Problem 441

Find the limit: $\lim \limits_{x\to -2}\dfrac {\sqrt[3]{x-6}+2}{x^{3}+8}$ .

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Problem 442

Find the limit: $\lim \limits_{x\to 16}\dfrac {\sqrt[4]{x}-2}{\sqrt{x}-4}$ .

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Problem 443

Find the limit: $\lim \limits_{x\to 8}\dfrac {\sqrt{9+2x}-5}{\sqrt[3]{x}-2}$ .

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Problem 444

Find the limit: $\lim \limits_{x\to 0}\dfrac {\sqrt[n]{1+x}-1}{x}$ ( $n$ is an integer).

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Problem 445

Find the limit: $\lim \limits_{x\to 0}\dfrac {\sqrt{1-2x-x^{2}}-(1+x)}{x}$ .

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Problem 446

Find the limit: $\lim \limits_{x\to 0}\dfrac {\sqrt[3]{8+3x-x^{2}}-2}{x+x^{2}}$ .

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Problem 447

Find the limit: $\lim \limits_{x\to 0}\dfrac {\sqrt[3]{27+x}-\sqrt[3]{27-x}}{x+2\sqrt[3]{x^{4}}}$ .

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Problem 448

Find the limit: $\lim \limits_{x\to 0}\dfrac {\sqrt{1+x}-\sqrt{1-x}}{\sqrt[3]{1+x}-\sqrt[3]{1-x}}$ .

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Problem 449

Find the limit: $\lim \limits_{x\to 7}\dfrac {\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}$ .

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Problem 450

Find the limit: $\lim \limits_{x\to 0}\dfrac {\sqrt[3]{1+\dfrac {x}{3}}-\sqrt[4]{1+\dfrac {x}{4}}}{1-\sqrt{1-\dfrac {x}{2}}}$ .

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Problem 451

Find the limit: $\lim \limits_{x\to 0}\dfrac {x^{2}}{\sqrt[5]{1+5x}-(1+x)}$ .

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Problem 452

Find the limit: $\lim \limits_{x\to 0}\dfrac {\sqrt[n]{1+\alpha x}-\sqrt[m]{1+\beta x}}{x}$ ( $m$ and $n$ are integers).

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Problem 453

Find the limit: $\lim \limits_{x\to 0}\dfrac {\sqrt[n]{1+\alpha x}\cdot \sqrt[m]{1+\beta x}-1}{x}$ ( $m$ and $n$ are integers).

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Problem 454

Let $P(x)=a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n}$ , and let $m$ denote an integer. Prove that:

\lim _{x\to 0}\frac{\sqrt[m]{1+P(x)}-1}{x}=\frac{a_{1}}{m} .

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Problem 455

Find the following limit: $\lim \limits_{x\to 1}\dfrac {\sqrt[m]{x}-1}{\sqrt[n]{x}-1}$ ( $m$ and $n$ denote integers).

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Problem 455.1

[NO TASK]

Problem 456

Find the following limit: $\lim \limits_{x\to 1}\dfrac {(1-\sqrt{x})(1-\sqrt[3]{x})\cdots (1-\sqrt[n]{x})}{(1-x)^{n-1}}$ .

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Problem 457

Find the limit: $\lim \limits_{x\to +\infty }\left[\sqrt{(x+a)(x+b)}-x\right]$ .

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Problem 458

Find the limit: $\lim \limits_{x\to +\infty }\left(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\right)$ .

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Problem 459

Find the limit: $\lim \limits_{x\to +\infty }x\left(\sqrt{x^{2}+2x}-2\sqrt{x^{2}+x}+x\right)$ .

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Problem 460

Find the limit: $\lim \limits_{x\to +0}\left[\sqrt{\dfrac {1}{x}+\sqrt{\dfrac {1}{x}+\sqrt{\dfrac {1}{x}}}}-\sqrt{\dfrac {1}{x}-\sqrt{\dfrac {1}{x}+\sqrt{\dfrac {1}{x}}}}\right]$ .

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Problem 461

Find the limit: $\lim \limits_{x\to \infty }\left(\sqrt[3]{x^{3}+x^{2}+1}-\sqrt[3]{x^{3}-x^{2}+1}\right)$ .

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Problem 462

Find the limit: $\lim \limits_{x\to +\infty }\left(\sqrt[3]{x^{3}+3x^{2}}-\sqrt{x^{2}-2x}\right)$ .

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Problem 463

Find the limit: $\lim \limits_{x\to \infty }x^{\frac{1}{3}}\left[(x+1)^{\frac{2}{3}}-(x-1)^{\frac{2}{3}}\right]$ .

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Problem 464

Find the limit: $\lim \limits_{x\to +\infty }x^{\frac{3}{2}}\left(\sqrt{x+2}-2\sqrt{x+1}+\sqrt{x}\right)$ .

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Problem 465

Find the limit: $\lim \limits_{x\to +\infty }\left[\sqrt[n]{(x+a_{1})\cdots (x+a_{n})}-x\right]$ .

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Problem 466

Find the limit: $\lim \limits_{x\to \infty }\dfrac {(x-\sqrt{x^{2}-1})^{n}+(x+\sqrt{x^{2}-1})^{n}}{x^{n}}$ ( $n$ denotes a natural number).

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Problem 467

Find the limit: $\lim \limits_{x\to 0}\dfrac {(\sqrt{1+x^{2}}+x)^{n}-(\sqrt{1+x^{2}}-x)^{n}}{x}$ ( $n$ denotes a natural number).

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Problem 468

Suppose in the quadratic equation $ax^{2}+bx+c=0$ the coefficient $a$ tends to zero while the coefficients $b$ and $c$ are constants, with $b\neq 0$ ; investigate the behavior of the two roots $x_{1}$ and $x_{2}$ of this quadratic equation.

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Problem 469

From the condition:

\lim _{x\to \infty }\left(\frac{x^{2}+1}{x+1}-ax-b\right)=0

find the constants $a$ and $b$ .

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Problem 470

From the conditions:

\lim _{x\to -\infty }\left(\sqrt{x^{2}-x+1}-a_{1}x-b_{1}\right)=0

and

\lim _{x\to +\infty }\left(\sqrt{x^{2}-x+1}-a_{2}x-b_{2}\right)=0

find the constants $a_{i}$ and $b_{i}$ ( $i=1,2$ ).

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Problem 471

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\sin 5x}{x}$ .

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Problem 472

Find the following limit: $\lim \limits_{x\to \infty }\dfrac {\sin x}{x}$ .

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Problem 473

Find the following limit: $\lim \limits_{x\to \pi }\dfrac {\sin mx}{\sin nx}$ ( $m$ and $n$ are integers).

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Problem 474

Find the following limit: $\lim \limits_{x\to 0}\dfrac {1-\cos x}{x^{2}}$ .

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Problem 474.1

[NO TASK]

Problem 474.2

[NO TASK]

Problem 475

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\tg x-\sin x}{\sin^{3}x}$ .

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Problem 476

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\sin 5x-\sin 3x}{\sin x}$ .

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Problem 477

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\cos x-\cos 3x}{x^{2}}$ .

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Problem 478

Find the following limit: $\lim \limits_{x\to 0}\dfrac {1+\sin x-\cos x}{1+\sin px-\cos px}$ .

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Problem 479

Find the following limit: $\lim \limits_{x\to \frac{\pi }{4}}\tg 2x\tg \left(\dfrac {\pi }{4}-x\right)$ .

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Problem 480

Find the following limit: $\lim \limits_{x\to 1}(1-x)\tg \dfrac {\pi x}{2}$ .

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Problem 481

(a)(b)(c)

Prove the identities:

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(a)

$\lim \limits_{x\to a}\sin x=\sin a$ ;

(b)

$\lim \limits_{x\to a}\cos x=\cos a$ ;

(c)

$\lim \limits_{x\to a}\tg x=\tg a$ ( $a\neq \dfrac {2n-1}{2}\pi ; n=0,\pm 1,\pm 2,\cdots$ ).

Problem 482

Find the following limit: $\lim \limits_{x\to a}\dfrac {\sin x-\sin a}{x-a}$ .

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Problem 483

Find the following limit: $\lim \limits_{x\to a}\dfrac {\cos x-\cos a}{x-a}$ .

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Problem 484

Find the following limit: $\lim \limits_{x\to a}\dfrac {\tg x-\tg a}{x-a}$ .

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Problem 485

Find the following limit: $\lim \limits_{x\to a}\dfrac {\ctg x-\ctg a}{x-a}$ .

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Problem 486

Find the following limit: $\lim \limits_{x\to a}\dfrac {\sec x-\sec a}{x-a}$ .

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Problem 487

Find the following limit: $\lim \limits_{x\to a}\dfrac {\csc x-\csc a}{x-a}$ .

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Problem 488

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\sin (a+2x)-2\sin (a+x)+\sin a}{x^{2}}$ .

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Problem 489

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\cos (a+2x)-2\cos (a+x)+\cos a}{x^{2}}$ .

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Problem 490

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\tg (a+2x)-2\tg (a+x)+\tg a}{x^{2}}$ .

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Problem 491

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\ctg (a+2x)-2\ctg (a+x)+\ctg a}{x^{2}}$ .

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Problem 492

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\sin (a+x)\sin (a+2x)-\sin^{2}a}{x}$ .

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Problem 493

Find the following limit: $\lim \limits_{x\to \frac{\pi }{6}}\dfrac {2\sin^{2}x+\sin x-1}{2\sin^{2}x-3\sin x+1}$ .

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Problem 494

Find the following limit: $\lim \limits_{x\to 0}\dfrac {1-\cos x\cos 2x\cos 3x}{1-\cos x}$ .

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Problem 495

Find the following limit: $\lim \limits_{x\to \frac{\pi }{3}}\dfrac {\sin \left(x-\dfrac {\pi }{3}\right)}{1-2\cos x}$ .

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Problem 496

Find the following limit: $\lim \limits_{x\to \frac{\pi }{3}}\dfrac {\tg^{3}x-3\tg x}{\cos \left(x+\dfrac {\pi }{6}\right)}$ .

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Problem 497

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\tg (a+x)\tg (a-x)-\tg^{2}a}{x^{2}}$ .

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Problem 498

Find the following limit: $\lim \limits_{x\to \frac{\pi }{4}}\dfrac {1-\ctg^{3}x}{2-\ctg x-\ctg^{3}x}$ .

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Problem 499

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\sqrt{1+\tg x}-\sqrt{1+\sin x}}{x^{3}}$ .

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Problem 500

Find the following limit: $\lim \limits_{x\to 0}\dfrac {x^{2}}{\sqrt{1+x\sin x}-\sqrt{\cos x}}$ .

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Problem 501

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\sqrt{\cos x}-\sqrt[3]{\cos x}}{\sin^{2}x}$ .

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Problem 502

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\sqrt{1-\cos x^{2}}}{1-\cos x}$ .

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Problem 503

Find the following limit: $\lim \limits_{x\to 0}\dfrac {1-\sqrt{\cos x}}{1-\cos (\sqrt{x})}$ .

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Problem 504

Find the following limit: $\lim \limits_{x\to 0}\dfrac {1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{x^{2}}$ .

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Problem 505

Find the following limit: $\lim \limits_{x\to +\infty }\left(\sin \sqrt{x+1}-\sin \sqrt{x}\right)$ .

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Problem 506

(a)(b)(c)

Find the following limits:

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(a)

$\lim \limits_{x\to 0}\left(\dfrac {1+x}{2+x}\right)^{\frac{1-\sqrt{x}}{1-x}}$ ;

(b)

$\lim \limits_{x\to 1}\left(\dfrac {1+x}{2+x}\right)^{\frac{1-\sqrt{x}}{1-x}}$ ;

(c)

$\lim \limits_{x\to +\infty }\left(\dfrac {1+x}{2+x}\right)^{\frac{1-\sqrt{x}}{1-x}}$ .

Problem 507

Find the limit: $\lim \limits_{x\to \infty }\left(\dfrac {x+2}{2x-1}\right)^{x^{2}}$ .

?

Problem 508

Find the limit: $\lim \limits_{x\to \infty }\left(\dfrac {3x^{2}-x+1}{2x^{2}+x+1}\right)^{\frac{x^{3}}{1-x}}$ .

?

Problem 509

Find the limit: $\lim \limits_{n\to \infty }\sin^{n}\dfrac {2\pi n}{3n+1}$ .

?

Problem 510

Find the limit: $\lim \limits_{x\to \frac{\pi }{4}+0}\left[\tg \left(\dfrac {\pi }{8}+x\right)\right]^{\tg 2x}$ .

?

Problem 511

Find the limit: $\lim \limits_{x\to \infty }\left(\dfrac {x^{2}-1}{x^{2}+1}\right)^{\frac{x-1}{x+1}}$ .

?

Problem 512

Find the limit: $\lim \limits_{x\to \infty }\left(\dfrac {x^{2}+1}{x^{2}-1}\right)^{x^{2}}$ .

?

Problem 513

Find the limit: $\lim \limits_{x\to \infty }\left(\dfrac {x^{2}+2x-1}{2x^{2}-3x-2}\right)^{\frac{1}{x}}$ .

?

Problem 514

Find the limit: $\lim \limits_{x\to 0}\sqrt[x]{1-2x}$ .

?

Problem 515

Find the limit: $\lim \limits_{x\to \infty }\left(\dfrac {x+a}{x-a}\right)^{x}$ .

?

Problem 516

(a)(b)(c)

Find the limit: $\lim \limits_{x\to +\infty }\left(\dfrac {a_{1}x+b_{1}}{a_{2}x+b_{2}}\right)^{x}$ ( $a_{1}>0,a_{2}>0$ ).

?

(a)

(b)

(c)

Problem 517

Find the limit: $\lim \limits_{x\to 0}(1+x^{2})^{\ctg^{2}x}$ .

?

Problem 518

Find the limit: $\lim \limits_{x\to 1}(1+\sin \pi x)^{\ctg \pi x}$ .

?

Problem 519

Find the limit: $\lim \limits_{x\to 0}\left(\dfrac {1+\tg x}{1+\sin x}\right)^{\frac{1}{\sin x}}$ .

?

Problem 519.1

[NO TASK]

Problem 520

Find the limit: $\lim \limits_{x\to a}\left(\dfrac {\sin x}{\sin a}\right)^{\frac{1}{x-a}}$ .

?

Problem 521

Find the limit: $\lim \limits_{x\to 0}\left(\dfrac {\cos x}{\cos 2x}\right)^{\frac{1}{x^{2}}}$ .

?

Problem 522

Find the limit: $\lim \limits_{x\to \frac{\pi }{4}}(\tg x)^{\tg 2x}$ .

?

Problem 523

Find the limit: $\lim \limits_{x\to \frac{\pi }{2}}(\sin x)^{\tg x}$ .

?

Problem 524

Find the limit: $\lim \limits_{x\to 0}\left[\tg \left(\dfrac {\pi }{4}-x\right)\right]^{\ctg x}$ .

?

Problem 525

Find the limit: $\lim \limits_{x\to \infty }\left(\sin \dfrac {1}{x}+\cos \dfrac {1}{x}\right)^{x}$ .

?

Problem 526

Find the limit: $\lim \limits_{x\to +0}\sqrt[x]{\cos \sqrt{x}}$ .

?

Problem 527

Find the limit: $\lim \limits_{n\to \infty }\left(\dfrac {n+x}{n-1}\right)^{n}$ .

?

Problem 528

Find the limit: $\lim \limits_{n\to \infty }\cos^{n}\dfrac {x}{\sqrt{n}}$ .

?

Problem 529

Find the limit: $\lim \limits_{x\to 0}\dfrac {\ln (1+x)}{x}$ .

?

Problem 530

Find the limit: $\lim \limits_{x\to +\infty }x\left[\ln (x+1)-\ln x\right]$ .

?

Problem 531

Find the limit: $\lim \limits_{x\to a}\dfrac {\ln x-\ln a}{x-a}$ ( $a>0$ ).

?

Problem 532

Find the limit: $\lim \limits_{x\to +\infty }\left[\sin \ln (x+1)-\sin \ln x\right]$ .

?

Problem 533

Find the limit: $\lim \limits_{x\to +\infty }\dfrac {\ln (x^{2}-x+1)}{\ln (x^{10}+x+1)}$ .

?

Problem 534

Find the limit: $\lim \limits_{x\to \infty }\left(\lg \dfrac {100+x^{2}}{1+100x^{2}}\right)$ .

?

Problem 535

Find the limit: $\lim \limits_{x\to +\infty }\dfrac {\ln (2+e^{3x})}{\ln (3+e^{2x})}$ .

?

Problem 536

Find the limit: $\lim \limits_{x\to +\infty }\dfrac {\ln (1+\sqrt{x}+\sqrt[3]{x})}{\ln (1+\sqrt[3]{x}+\sqrt[4]{x})}$ .

?

Problem 537

Find the limit: $\lim \limits_{h\to 0}\dfrac {\log (x+h)+\log (x-h)-2\log x}{h^{2}}$ ( $x>0$ ).

?

Problem 538

Find the limit: $\lim \limits_{x\to 0}\dfrac {\ln \tg \left(\dfrac {\pi }{4}+ax\right)}{\sin bx}$ .

?

Problem 539

Find the limit: $\lim \limits_{x\to 0}\dfrac {\ln \cos ax}{\ln \cos bx}$ .

?

Problem 540

Find the limit: $\lim \limits_{x\to 0}\left[\ln \dfrac {nx+\sqrt{1-n^{2}x^{2}}}{x+\sqrt{1-x^{2}}}\right]$ .

?

Problem 540.1

[NO TASK]

Problem 541

Find the limit: $\lim \limits_{x\to 0}\dfrac {a^{x}-1}{x}$ ( $a>0$ ).

?

Problem 542

Find the limit: $\lim \limits_{x\to a}\dfrac {a^{x}-x^{a}}{x-a}$ ( $a>0$ ).

?

Problem 543

Find the limit: $\lim \limits_{x\to a}\dfrac {x^{x}-a^{a}}{x-a}$ ( $a>0$ ).

?

Problem 544

Find the limit: $\lim \limits_{x\to 0}(x+e^{x})^{\frac{1}{x}}$ .

?

Problem 545

Find the limit: $\lim \limits_{x\to 0}\left(\dfrac {1+x\cdot 2^{x}}{1+x\cdot 3^{x}}\right)^{\frac{1}{x^{2}}}$ .

?

Problem 545.1

[NO TASK]

Problem 545.2

[NO TASK]

Problem 545.3

[NO TASK]

Problem 546

Find the limit: $\lim \limits_{n\to \infty }\tg^{n}\left(\dfrac {\pi }{4}+\dfrac {1}{n}\right)$ .

?

Problem 547

Find the limit: $\lim \limits_{x\to 0}\dfrac {e^{\alpha x}-e^{\beta x}}{\sin \alpha x-\sin \beta x}$ .

?

Problem 548

Find the limit: $\lim \limits_{x\to a}\dfrac {x^{\alpha }-a^{\alpha }}{x^{\beta }-a^{\beta }}$ ( $a>0$ ).

?

Problem 549

Find the limit: $\lim \limits_{x\to b}\dfrac {a^{x}-a^{b}}{x-b}$ ( $a>0$ ).

?

Problem 550

Find the limit: $\lim \limits_{h\to 0}\dfrac {a^{x+h}+a^{x-h}-2a^{x}}{h^{2}}$ ( $a>0$ ).

?

Problem 551

Find the limit: $\lim \limits_{x\to \infty }\dfrac {(x+a)^{x+a}(x+b)^{x+b}}{(x+a+b)^{2x+a+b}}$ .

?

Problem 552

Find the limit: $\lim \limits_{n\to \infty }n\left(\sqrt[n]{x}-1\right)$ ( $x>0$ ).

?

Problem 553

Find the limit: $\lim \limits_{n\to \infty }n^{2}\left(\sqrt[n]{x}-\sqrt[n+1]{x}\right)$ ( $x>0$ ).

?

Problem 554

Find the limit: $\lim \limits_{n\to \infty }\left(\dfrac {a-1+\sqrt[n]{b}}{a}\right)^{n}$ ( $a>0,b>0$ ).

?

Problem 555

Find the limit: $\lim \limits_{n\to \infty }\left(\dfrac {\sqrt[n]{a}+\sqrt[n]{b}}{2}\right)^{n}$ ( $a>0,b>0$ ).

?

Problem 556

Find the limit: $\lim \limits_{x\to 0}\left(\dfrac {a^{x}+b^{x}+c^{x}}{3}\right)^{\frac{1}{x}}$ ( $a>0,b>0,c>0$ ).

?

Problem 557

Find the limit: $\lim \limits_{x\to 0}\left(\dfrac {a^{x+1}+b^{x+1}+c^{x+1}}{a+b+c}\right)^{\frac{1}{x}}$ ( $a>0,b>0,c>0$ ).

?

Problem 558

Find the limit: $\lim \limits_{x\to 0}\left(\dfrac {a^{x^{2}}-b^{x^{2}}}{a^{x}+b^{x}}\right)^{\frac{1}{x}}$ ( $a>0,b>0$ ).

?

Problem 559

Find the limit: $\lim \limits_{x\to 0}\dfrac {a^{x^{2}}-b^{x^{2}}}{(a^{x}-b^{x})^{2}}$ ( $a>0,b>0$ ).

?

Problem 560

Find the limit: $\lim \limits_{x\to a}\dfrac {a^{a^{x}}-a^{x^{a}}}{a^{x}-x^{a}}$ ( $a>0$ ).

?

Problem 561

(a)(b)

Find the following limits:

?

(a)

$\lim \limits_{x\to -\infty }\dfrac {\ln (1+3^{x})}{\ln (1+2^{x})}$ ;

(b)

$\lim \limits_{x\to +\infty }\dfrac {\ln (1+3^{x})}{\ln (1+2^{x})}$ .

Problem 562

Find the limit: $\lim \limits_{x\to +\infty }\ln (1+2^{x})\ln \left(1+\dfrac {3}{x}\right)$ .

?

Problem 563

Find the limit: $\lim \limits_{x\to 1}(1-x)\lg_{x}2$ .

?

Problem 564

Prove that:

\lim _{x\to +\infty }\frac{x^{n}}{a^{x}}=0 \quad (a>1,n>0) .

?

Problem 565

Prove that:

\lim _{x\to +\infty }\frac{\lg _{a}x}{x^{\varepsilon }}=0 \quad (a>1,\varepsilon >0) .

?

Problem 566

(a)(b)

Find the following limits:

?

(a)

$\lim \limits_{x\to 0}\dfrac {\ln (x^{2}+e^{x})}{\ln (x^{4}+e^{2x})}$ ;

(b)

$\lim \limits_{x\to +\infty }\dfrac {\ln (x^{2}+e^{x})}{\ln (x^{4}+e^{2x})}$ .

Problem 567

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\ln (1+xe^{x})}{\ln (x+\sqrt{1+x^{2}})}$ .

?

Problem 568

Find the following limit: $\lim \limits_{x\to \infty }\left[(x+2)\ln (x+2)-2(x+1)\ln (x+1)+x\ln x\right]$ .

?

Problem 569

Find the following limit: $\lim \limits_{x\to +0}\left[\ln (x\ln a)\cdot \ln \left(\dfrac {\ln ax}{\ln \dfrac {x}{a}}\right)\right]$ ( $a>1$ ).

?

Problem 570

Find the following limit: $\lim \limits_{x\to +\infty }\left[\ln \dfrac {x+\sqrt{x^{2}+1}}{x+\sqrt{x^{2}-1}}\cdot \ln^{-2}\dfrac {x+1}{x-1}\right]$ .

?

Problem 571

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\sqrt{1+x\sin x}-1}{e^{x^{2}}-1}$ .

?

Problem 572

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\cos (xe^{x})-\cos (xe^{-x})}{x^{3}}$ .

?

Problem 573

Find the following limit: $\lim \limits_{x\to 0}\left(2e^{\frac{x}{x+1}}-1\right)^{\frac{x^{2}+1}{x}}$ .

?

Problem 574

Find the following limit: $\lim \limits_{x\to 1}(2-x)^{\sec \frac{\pi x}{2}}$ .

?

Problem 575

Find the following limit: $\lim \limits_{x\to \frac{\pi }{2}}\dfrac {1-\sin^{\alpha +\beta }x}{\sqrt{(1-\sin^{\alpha }x)(1-\sin^{\beta }x)}}$ ( $\alpha >0,\beta >0$ ).

?

Problem 576

(a)(b)(c)

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\sh^{2}x}{\ln (\ch 3x)}$ ( $\sh x=\dfrac {e^{x}-e^{-x}}{2}$ , $\ch x=\dfrac {e^{x}+e^{-x}}{2}$ , cf. Problem 340).

?

(a)

(b)

(c)

Problem 576.1

[NO TASK]

Problem 577

Find the following limit: $\lim \limits_{x\to +\infty }\dfrac {\sh \sqrt{x^{2}+x}-\sh \sqrt{x^{2}-x}}{\ch x}$ .

?

Problem 577.1

(a)(b)

(a)

(b)

Problem 577.2

[NO TASK]

Problem 578

Find the following limit: $\lim \limits_{x\to +\infty }(x-\ln \ch x)$ .

?

Problem 579

Find the following limit: $\lim \limits_{x\to 0}\dfrac {e^{\sin 2x}-e^{\sin x}}{\th x}$ .

?

Problem 580

Find the following limit: $\lim \limits_{n\to \infty }\left(\dfrac {\ch \dfrac {\pi }{n}}{\cos \dfrac {\pi }{n}}\right)^{n^{2}}$ .

?

Problem 581

Find the following limit: $\lim \limits_{x\to \infty }\arcsin \dfrac {1-x}{1+x}$ .

?

Problem 582

Find the following limit: $\lim \limits_{x\to +\infty }\arccos \left(\sqrt{x^{2}+x}-x\right)$ .

?

Problem 583

Find the following limit: $\lim \limits_{x\to 2}\arcctg \dfrac {x-4}{(x-2)^{2}}$ .

?

Problem 584

Find the following limit: $\lim \limits_{x\to -\infty }\arcctg \dfrac {x}{\sqrt{1+x^{2}}}$ .

?

Problem 585

Find the following limit: $\lim \limits_{h\to 0}\dfrac {\arctg (x+h)-\arctg x}{h}$ .

?

Problem 586

Find the following limit: $\lim \limits_{x\to 0}\dfrac {\ln \dfrac {1+x}{1-x}}{\arctg (1+x)-\arctg (1-x)}$ .

?

Problem 587

Find the following limit: $\lim \limits_{n\to \infty }\left[n\arctg \dfrac {1}{n(x^{2}+1)+x}\cdot \tg^{n}\left(\dfrac {\pi }{4}+\dfrac {x}{2n}\right)\right]$ .

?

Problem 588

Find the following limit: $\lim \limits_{x\to \infty }x\left(\dfrac {\pi }{4}-\arctg \dfrac {x}{x+1}\right)$ .

?

Problem 589

Find the following limit: $\lim \limits_{x\to +\infty }x\left(\dfrac {\pi }{2}-\arcsin \dfrac {x}{\sqrt{x^{2}+1}}\right)$ .

?

Problem 590

Find the following limit: $\lim \limits_{n\to \infty }\left[1+\dfrac {(-1)^{n}}{n}\right]^{\csc \left(\pi \sqrt{1+n^{2}}\right)}$ .

?

Problem 591

Find the following limit: $\lim \limits_{x\to 0}\dfrac {1}{x^{100}}e^{-\frac{1}{x^{2}}}$ .

?

Problem 592

Find the following limit: $\lim \limits_{x\to +0}x\ln x$ .

?

Problem 593

(a)(b)

Find the following limits:

?

(a)

$\lim \limits_{x\to -\infty }\left(\sqrt{x^{2}+x}-x\right)$ ;

(b)

$\lim \limits_{x\to +\infty }\left(\sqrt{x^{2}+x}-x\right)$ .

Problem 594

(a)(b)

Find the following limits:

?

(a)

(b)

$\lim \limits_{x\to +\infty }\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)$ .

Problem 594.1

$\lim \limits_{x\to -\infty }\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)$ ;

?

Problem 595

(a)(b)

Find the following limits:

?

(a)

$\lim \limits_{x\to 1-0}\arctg \dfrac {1}{1-x}$ ;

(b)

$\lim \limits_{x\to 1+0}\arctg \dfrac {1}{1-x}$ .

Problem 596

(a)(b)

Find the following limits:

?

(a)

$\lim \limits_{x\to -0}\dfrac {1}{1+e^{\frac{1}{x}}}$ ;

(b)

$\lim \limits_{x\to +0}\dfrac {1}{1+e^{\frac{1}{x}}}$ .

Problem 597

(a)(b)

Find the following limits:

?

(a)

$\lim \limits_{x\to -\infty }\dfrac {\ln (1+e^{x})}{x}$ ;

(b)

$\lim \limits_{x\to +\infty }\dfrac {\ln (1+e^{x})}{x}$ .

Problem 598

(a)(b)

Prove:

?

(a)

as $x\to -\infty$ , $\dfrac {2x}{1+x}\to 2+0$ ;

(b)

as $x\to +\infty$ , $\dfrac {2x}{1+x}\to 2-0$ .

Problem 599

(a)(b)

Prove:

?

(a)

as $x\to -0$ , $2^{x}\to 1-0$ ;

(b)

as $x\to +0$ , $2^{x}\to 1+0$ .

Problem 600

Let $f(x)=x+\left[x^{2}\right]$ , find $f(1)$ , $f(1-0)$ , $f(1+0)$ .

?

Problem 601

Let $f(x)=\operatorname {sign}(\sin \pi x)$ , find $f(n)$ , $f(n-0)$ , $f(n+0)$ ( $n=0,\pm 1,\cdots$ ).

?

Problem 602

Find: $\lim \limits_{x\to 0}x\sqrt{\cos \dfrac {1}{x}}$ .

?

Problem 603

Find: $\lim \limits_{x\to 0}x\left[\dfrac {1}{x}\right]$ .

?

Problem 604

Find: $\lim \limits_{n\to \infty }\sin \left(\pi \sqrt{n^{2}+1}\right)$ .

?

Problem 605

Find: $\lim \limits_{n\to \infty }\sin^{2}\left(\pi \sqrt{n^{2}+n}\right)$ .

?

Problem 606

Find: $\lim \limits_{n\to \infty }\underbrace{\sin \sin \cdots \sin }_{n\uparrow }x$ .

?

Problem 607

Let $\lim \limits_{x\to a}\varphi (x)=A$ and $\lim \limits_{x\to A}\psi (x)=B$ . Does it follow that

\lim _{x\to a}\psi (\varphi (x))=B?

Study this example: when $x=\dfrac {p}{q}$ (where $p$ and $q$ are coprime integers), $\varphi (x)=\dfrac {1}{q}$ ; when $x$ is irrational, $\varphi (x)=0$ ; when $x\neq 0$ , $\psi (x)=1$ ; when $x=0$ , $\psi (x)=0$ ; and let $x\to 0$ .

?

Problem 608

(a)(b)

Prove Cauchy's theorem: if the function $f(x)$ is defined on the interval $(a,+\infty )$ , and is bounded on every finite interval $(a,b)$ , then

?

(a)

$\lim \limits_{x\to +\infty }\dfrac {f(x)}{x}=\lim \limits_{x\to +\infty }\left[f(x+1)-f(x)\right]$ ;

(b)

$\lim \limits_{x\to +\infty }\left[f(x)\right]^{\frac{1}{x}}=\lim \limits_{x\to +\infty }\dfrac {f(x+1)}{f(x)}$ ( $f(x)\geqslant C>0$ ),

assuming the limits on the right-hand side of the equalities exist.

Problem 609

Prove: if (1) the function $f(x)$ is defined on the domain $x>a$ ; (2) it is bounded on every finite domain $a<x<b$ ; (3) $\lim \limits_{x\to +\infty }\left[f(x+1)-f(x)\right]=+\infty$ (or $-\infty$ ), then $\lim \limits_{x\to +\infty }\dfrac {f(x)}{x}=+\infty (-\infty )$ .

Note: the original problem should be corrected to read: if (1) $f(x)$ is defined on the domain $x>a$ , (2) it is bounded on every finite domain $a<x<b$ , (3) $\lim \limits_{x\to +\infty }\left[f(x+1)-f(x)\right]=+\infty$ (or $-\infty$ ), then $\lim \limits_{x\to +\infty }\dfrac {f(x)}{x}=+\infty (-\infty )$ .

In the original problem condition (3) was mistakenly given as $\lim \limits_{x\to +\infty }\left[f(x+1)-f(x)\right]=\infty$ , and the conclusion was mistakenly given as $\lim \limits_{x\to +\infty }\dfrac {f(x)}{x}=\infty$ . For example, define the function $f(x)$ on $\left[0,+\infty \right]$ by

f(x)= \begin{cases} 2n, & x\in \left[2n,2n+1\right] ; \\ -2n, & x\in \left[2n+1,2n+2\right] , \end{cases} \quad n=0,1,2,\cdots

Clearly $f(x)$ satisfies conditions (1), (2), and

\lim _{x\to +\infty }\left[f(x+1)-f(x)\right]=\infty .

But

\lim _{x\to +\infty }\frac{f(x)}{x}\neq \infty ,

indeed

\varlimsup _{x\to +\infty }\frac{f(x)}{x}=-1 ,\quad \overline{\lim _{x\to +\infty }}\frac{f(x)}{x}=1 .

?

Problem 610

Prove that, if: (1) the function $f(x)$ is defined on the domain $x>a$ ; (2) it is bounded on every finite domain $a<x<b$ ; (3) there exists a finite or infinite (with definite sign, i.e., $+\infty$ or $-\infty$ ) limit

\lim _{x\to +\infty }\frac{f(x+1)-f(x)}{x^{n}}=l ,

then

\lim _{x\to \infty }\frac{f(x)}{x^{n+1}}=\frac{l}{n+1} .

?

Problem 611

(a)(b)

Prove:

?

(a)

$\lim \limits_{n\to \infty }\left(1+\dfrac {x}{n}\right)^{n}=e^{x}$ ;

(b)

$\lim \limits_{n\to \infty }\left(1+x+\dfrac {x^{2}}{2!}+\cdots +\dfrac {x^{n}}{n!}\right)=e^{x}$ .

Problem 612

Prove: $\lim \limits_{n\to \infty }n\sin (2\pi e\cdot n!)=2\pi$ .

?

Problem 613

(a)(b)

Sketch the graph of the following function:

?

(a)

$y=1-x^{100}$ ;

(b)

$y=\lim \limits_{n\to \infty }(1-x^{2n})$ ( $-1\leqslant x\leqslant 1$ ).

Problem 614

(a)(b)

Sketch the graph of the following function:

?

(a)

$y=\dfrac {x^{100}}{1+x^{100}}$ ( $x\geqslant 0$ );

(b)

$y=\lim \limits_{n\to \infty }\dfrac {x^{n}}{1+x^{n}}$ ( $x\geqslant 0$ ).

Problem 615

Sketch the graph of the following function: $y=\lim \limits_{n\to \infty }\dfrac {x^{n}-x^{-n}}{x^{n}+x^{-n}}$ ( $x\neq 0$ ).

?

Problem 616

Sketch the graph of the following function: $y=\lim \limits_{n\to \infty }\sqrt[n]{x^{2}+\dfrac {1}{n^{2}}}$ .

?

Problem 617

Sketch the graph of the following function: $y=\lim \limits_{n\to \infty }\sqrt[n]{1+x^{n}}$ ( $x\geqslant 0$ ).

?

Problem 618

Sketch the graph of the following function: $y=\lim \limits_{n\to \infty }\sqrt[n]{1+x^{n}+\left(\dfrac {x^{2}}{2}\right)^{n}}$ ( $x\geqslant 0$ ).

?

Problem 619

Sketch the graph of the following function: $y=\lim \limits_{n\to \infty }\dfrac {x^{n+2}}{\sqrt{2^{2n}+x^{2n}}}$ ( $x\geqslant 0$ ).

?

Problem 620

(a)(b)

Sketch the graphs of the following functions:

?

(a)

$y=\sin^{1000}x$ ;

(b)

$y=\lim \limits_{n\to \infty }\sin^{2n}x$ .

Problem 621

Sketch the graph of the following function: $y=\lim \limits_{n\to \infty }\dfrac {\ln (2^{n}+x^{n})}{n}$ ( $x\geqslant 0$ ).

?

Problem 622

Sketch the graph of the following function: $y=\lim \limits_{n\to \infty }(x-1)\operatorname {arctg}x^{n}$ .

?

Problem 623

Sketch the graph of the following function: $y=\lim \limits_{n\to \infty }\sqrt[n]{1+e^{n(x+1)}}$ .

?

Problem 624

Sketch the graph of the following function: $y=\lim \limits_{t\to +\infty }\dfrac {x+e^{tx}}{1+e^{tx}}$ .

?

Problem 625

Sketch the graph of the following function: $y=\lim \limits_{t\to x}\dfrac {1}{t-x}\ln \dfrac {t}{x}$ ( $x>0$ ).

?

Problem 625.1

[NO TASK]

Problem 625.2

[NO TASK]

Problem 625.3

[NO TASK]

Problem 626

If

\lim _{x\to \infty }\left[f(x)-(kx+b)\right]=0 ,

then the line $y=kx+b$ is called a (slant) asymptote of the curve $y=f(x)$ . Use this equation to derive the necessary and sufficient condition for the existence of an asymptote.

?

Problem 627

Find the asymptotes of the following curves and sketch their graphs:

?

(a)

$y=\dfrac {x^{2}}{x^{2}+x-2}$ ;

(b)

$y=\sqrt{x^{2}+x}$ ;

(c)

$y=\sqrt[3]{x^{2}-x^{3}}$ ;

(d)

$y=\dfrac {xe^{x}}{e^{x}-1}$ ;

(e)

$y=\ln (1+e^{x})$ ;

(f)

$y=x+\operatorname {arccos}\dfrac {1}{x}$ ;

(g)

$y=\dfrac {x^{1+x}}{(1+x)^{x}}$ .

Problem 628

Find the following limit: $\lim \limits_{n\to \infty }\left(\dfrac {x^{n+1}}{(n+1)!}+\dfrac {x^{n+2}}{(n+2)!}+\cdots +\dfrac {x^{2n}}{(2n)!}\right)$ .

?

Problem 629

Find the following limit: $\lim \limits_{n\to \infty }\left[(1+x)(1+x^{2})(1+x^{4})\cdots (1+x^{2^{n}})\right]$ , if $\left|x\right|<1$ .

?

Problem 630

Find the following limit: $\lim \limits_{n\to \infty }\left(\cos \dfrac {x}{2}\cos \dfrac {x}{4}\cdots \cos \dfrac {x}{2^{n}}\right)$ .

?

Problem 631

Let

\lim _{x\to 0}\frac{\varphi (x)}{\psi (x)}=1 ,

where $\psi (x)>0$ and as $n\to \infty$ , $\alpha_{mn}\to 0$ ( $m=1,2,\ldots ,n$ ), in other words, when $m=1,2,\ldots ,n$ and $n>N(\varepsilon )$ we have $\left|\alpha_{mn}\right|<\varepsilon$ . Also assume $\alpha_{mn}\neq 0$ .

Editor's note: this problem should have the condition $\alpha_{mn}\neq 0$ added (absent in the original book), because $\varphi (x)$ or $\psi (x)$ may be undefined at $x=0$ . In addition, $m=1,2,\ldots ,n$ .

Prove:

\lim _{n\to \infty }\left[\varphi (\alpha _{1n})+\varphi (\alpha _{2n})+\cdots +\varphi (\alpha _{nn})\right]=\lim _{n\to \infty }\left[\psi (\alpha _{1n})+\psi (\alpha _{2n})+\cdots +\psi (\alpha _{nn})\right] ,

where it is assumed that the limit on the right-hand side of the equality exists.

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Problem 632

Using the theorem of problem 631, find $\lim \limits_{n\to \infty }\sum \limits_{k=1}^{n}\left[\sqrt[3]{1+\dfrac {k}{n^{2}}}-1\right]$ .

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Problem 633

Find the following limit: $\lim \limits_{n\to \infty }\sum \limits_{k=1}^{n}\left(\sin \dfrac {ka}{n^{2}}\right)$ .

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Problem 634

Find the following limit: $\lim \limits_{n\to \infty }\sum \limits_{k=1}^{n}\left(a^{\frac{k}{n^{2}}}-1\right)$ ( $a>0$ ).

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Problem 635

Find the following limit: $\lim \limits_{n\to \infty }\prod \limits_{k=1}^{n}\left(1+\dfrac {k}{n^{2}}\right)$ .

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Problem 636

Find the following limit: $\lim \limits_{n\to \infty }\prod \limits_{k=1}^{n}\cos \dfrac {ka}{n\sqrt{n}}$ .

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Problem 637

The sequence $x_{n}$ is given by the following equalities:

x_{1}=\sqrt{a},\quad x_{2}=\sqrt{a+\sqrt{a}},\quad x_{3}=\sqrt{a+\sqrt{a+\sqrt{a}}},\quad \ldots \quad (a>0) .

Find $\lim \limits_{n\to \infty }x_{n}$ .

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Problem 637.1

[NO TASK]

Problem 637.2

[NO TASK]

Problem 637.3

[NO TASK]

Problem 638

The function sequence

y_{n}=y_{n}(x)\quad (0\leqslant x\leqslant 1)

is determined by the following method:

y_{1}=\frac{x}{2},\quad y_{n}=\frac{x}{2}-\frac{y_{n-1}^{2}}{2}\quad (n=2,3,\ldots ) .

Find $\lim \limits_{n\to \infty }y_{n}$ .

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Problem 639

The function sequence

y_{n}=y_{n}(x)\quad (0\leqslant x\leqslant 1)

is determined by the following method:

y_{1}=\frac{x}{2},\quad y_{n}=\frac{x}{2}+\frac{y_{n-1}^{2}}{2}\quad (n=2,3,\ldots ) .

Find $\lim \limits_{n\to \infty }y_{n}$ .

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Problem 639.1

[NO TASK]

Problem 639.2

[NO TASK]

Problem 640

In order to find an approximate solution of Kepler's equation (Уравнение Кеплера)

x-\varepsilon \sin x=m\quad (0<\varepsilon <1) \tag {1}

suppose

x_{0}=m,\quad x_{1}=m+\varepsilon \sin x_{0},\quad \ldots ,\quad x_{n}=m+\varepsilon \sin x_{n-1},\quad \ldots \quad (\text{successive approximation method}).

Prove that $\xi =\lim \limits_{n\to \infty }x_{n}$ exists, and that the number $\xi$ is the unique root of equation (1).

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Problem 641

If $\omega_{k}(f)$ is the amplitude (oscillation) of the function $f(x)$ on the interval $\left|x-\xi \right|\leqslant k$ ( $k>0$ ), then the number

\omega _{0}(f)=\lim _{k\to 0}\omega _{k}(f)

is called the amplitude of the function $f(x)$ at the point $\xi$ .

Find the amplitude of the following functions $f(x)$ at the point $x=0$ :

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(a)

$f(x)=\sin \dfrac {1}{x}$ ;

(b)

$f(x)=\dfrac {1}{x^{2}}\cos^{2}\dfrac {1}{x}$ ;

(c)

$f(x)=x\left(2+\sin \dfrac {1}{x}\right)$ ;

(d)

$f(x)=\dfrac {1}{\pi }\operatorname {arctg}\dfrac {1}{x}$ ;

(e)

$f(x)=\dfrac {\left|\sin x\right|}{x}$ ;

(f)

$f(x)=\dfrac {1}{1+e^{\frac{1}{x}}}$ ;

(g)

$f(x)=(1+\left|x\right|)^{\frac{1}{x}}$ .

Problem 642

Let

f(x)=\sin \frac{1}{x} .

Prove that for any number $\alpha$ satisfying the condition $-1\leqslant \alpha \leqslant 1$ , one can choose a sequence $x_{n}\to 0$ ( $n=1,2,\ldots$ ) such that

\lim _{n\to \infty }f(x_{n})=\alpha .

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Problem 643

(a)(b)(c)

Let:

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(a)

$f(x)=\sin^{2}\dfrac {1}{x}+\dfrac {2}{\pi }\operatorname {arctg}\dfrac {1}{x}$ ;

(b)

$f(x)=(2-x^{2})\cos \dfrac {1}{x}$ ;

(c)

$f(x)=\left(1+\cos^{2}\dfrac {1}{x}\right)^{\sec^{2}\frac{1}{x}}$ .

Find $l=\varliminf \limits_{x\to 0}f(x)$ and $L=\varlimsup \limits_{x\to 0}f(x)$ .

Problem 644

(a)(b)(c)(d)

Let:

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(a)

$f(x)=\sin x$ ;

(b)

$f(x)=x^{2}\cos^{2}x$ ;

(c)

$f(x)=2^{\sin^{2}x}$ ;

(d)

$f(x)=\dfrac {x}{1+x^{2}\sin^{2}x}$ ( $x\geqslant 0$ ).

Find $l=\varliminf \limits_{x\to \infty }f(x)$ and $L=\varlimsup \limits_{x\to \infty }f(x)$ .