Theory of Limits

Find the limit $\lim_{x \to 0} \frac{x^{4}+3 x^{2}}{x^{5}+x^{3}+2 x^{2}}$.

Find the limit $\lim_{x \to 1} \frac{x^{4}+2 x^{2}-3}{x^{2}-3 x+2}$.

Find the limit $\lim_{x \to 1} \frac{3 x^{4}-4 x^{3}+1}{(x-1)^{2}}$.

Find the limit $\lim_{x \to 4} \frac{x^{2}-6 x+8}{x^{2}-5 x+4}$.

Find the limit $\lim_{x \to 1} \frac{x^{n}-1}{x-1}$; $n$ is an integer.

Find the limit $\lim_{x \to 1} \frac{x^{m}-1}{x^{n}-1}$; $m$ and $n$ are integers.

Find the limit $\lim_{x \to 1} \left(\frac{1}{1-x} - \frac{3}{1-x^{3}}\right)$.

Find the limit $\lim_{x \to 1} \left(\frac{a}{1-x^{a}} - \frac{b}{1-x^{b}}\right)$; $a$ and $b$ are integers.

Find the limit $\lim_{x \to 1} \frac{(x^{n}-1)(x^{n-1}-1) \ldots (x^{n-k+1}-1)}{(x-1)(x^{2}-1) \ldots (x^{k}-1)}$.

Find the limit $\lim_{x \to 1} \frac{x^{p/q}-1}{x^{r/s}-1}$.

Find the limit $\lim_{x \to 0} \frac{\sqrt[3]{1+x}-1}{x}$.

Find the limit $\lim_{x \to -1} \frac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$.

Find the limit $\lim_{x \to 0} \frac{\sqrt[n]{1+x}-1}{x}$.

Find the limit $\lim_{n \to \infty } \frac{n^{2}+n+1}{(n-1)^{2}}$.

Find the limit $\lim_{n \to \infty } \frac{(n+1)(n+2)(n+3)}{n^{4}+n^{2}+1}$.

Find the limit $\lim_{n \to \infty } \frac{\sqrt[3]{n^{2}+n}}{n+2}$.

Find the limit $\lim_{n \to \infty } \frac{\sqrt{n^{3}+1}+\sqrt{n}}{4 \sqrt{n^{2}+n}-n}$.

Find the limit $\lim_{x \to \pm \infty } \frac{\sqrt{x^{2}+1}}{x+1}$.

Find the limit $\lim_{x \to \pm \infty } \frac{\sqrt{x^{2}+3 x}}{\sqrt[3]{x^{3}-2 x^{2}}}$.

Find the limit $\lim_{n \to \infty } \left(\frac{1+2+3+\ldots +n}{n+2} - \frac{n}{2}\right)$.

Find the limit $\lim_{n \to \infty } \frac{1-2+3-\ldots -2 n}{\sqrt{n^{2}+1}}$.

Find the limit $\lim_{n \to \infty } \frac{1}{n} \left(\left(a+\frac{1}{n}\right)^{2} + \left(a+\frac{2}{n}\right)^{2} + \ldots + \left(a+\frac{n-1}{n}\right)^{2}\right)$.

Find the limit $\lim_{n \to \infty } \frac{1+2^{2}+3^{2}+\ldots +n^{2}}{n^{3}}$.

Find the limit $\lim_{n \to \infty } \left(\frac{1+2^{2}+\ldots +n^{2}}{n^{2}} - \frac{n}{3}\right)$.

Find the limit $\lim_{n \to \infty } \frac{1^{2}+3^{2}+5^{2}+\ldots +(2n-1)^{2}}{n^{3}}$.

Find the limit $\lim_{n \to \infty } \frac{1 \cdot 2+2 \cdot 3+\ldots +n(n+1)}{n^{3}}$.

Find the limit $\lim_{x \to 0} \frac{\sqrt{1+x+x^{2}}-1}{x}$.

Find the limit $\lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1+x^{2}}}{\sqrt{1+x}-1}$.

Find the limit $\lim_{x \to 0} \frac{\sqrt{3+x+x^{2}}-\sqrt{9-2 x+x^{2}}}{x^{2}-3 x+2}$.

Find the limit $\lim_{x \to 0} \frac{5 x}{\sqrt[3]{1+x}-\sqrt[3]{1-x}}$.

Find the limit $\lim_{x \to 0} \frac{\sqrt[3]{1+3 x^{2}}-1}{x^{2}+x^{3}}$.

Find the limit $\lim_{x \to -1} \frac{\sqrt[3]{1+2 x}+1}{\sqrt[3]{2+x}+x}$.

Find the limit $\lim_{x \to 0} \frac{\sqrt[3]{1+3 x}-\sqrt{1-2 x}}{x+x^{2}}$.

Find the limit $\lim_{x \to \pm \infty } \left(\sqrt{x^{2}+1}-1\right)$.

Find the limit $\lim_{x \to \infty } \left(\sqrt{x^{2}+1}-\sqrt{x^{2}-1}\right)$.

Find the limit $\lim_{x \to \infty } \left(\sqrt{x^{2}+x+1}-\sqrt{x^{2}-x+1}\right)$.

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

Find the limit $\lim_{x \to +\infty } x \left(\sqrt{x^{2}+1}-x\right)$.

Find the limit $\lim_{x \to \infty } \left((x+1)^{2/3}-(x-1)^{2/3}\right)$.

Find the limit $\lim_{x \to \infty } \left(x^{4/3}-(x^{2}-1)^{2/3}\right)$.

Find the limit $\lim_{x \to +\infty } x^{3/2} \left(\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x}\right)$.

Find the limit $\lim_{x \to +\infty } x^{3} \left(\sqrt{x^{2}+\sqrt{x^{4}+1}}-x\sqrt{2}\right)$.

Find the limit $\lim_{x \to +\infty } \left(\sqrt[n]{(x+a_{1})(x+a_{2}) \ldots (x+a_{n})}-x\right)$.

Find the limit $\lim_{x \to \infty } x^{4/3} \left(\sqrt[3]{x^{2}+1}-\sqrt[3]{x^{2}-1}\right)$.

Determine $\lambda$ and $\mu$ from the condition

Determine $\lambda$ and $\mu$ from the condition

Find the limit $\lim_{x \to 0} \frac{\sqrt[n]{a+x}-\sqrt[n]{a-x}}{x}$ ($n$ is a positive integer).

Find the limit $\lim_{x \to 0} \frac{\sqrt{1+3x}+\sqrt[3]{1+x}-\sqrt[5]{1+x}-\sqrt[7]{1+x}}{\sqrt[4]{1+2x}+x-\sqrt[6]{1+x}}$.

Find the limit $\lim_{x \to 0} \frac{\sqrt[3]{a^{2}+a x+x^{2}}-\sqrt[3]{a^{2}-a x+x^{2}}}{\sqrt{a+x}-\sqrt{a-x}}$.

Find the limit $\lim_{x \to 0} \frac{\left(\sqrt{1+x^{2}}+x\right)^{n}-\left(\sqrt{1+x^{2}}-x\right)^{n}}{x}$.

Find the limit $\lim_{n \to \infty } \frac{a^{n}}{1+a^{n}}$; $a > 0$.

Find the limit $\lim_{n \to \infty } \frac{a^{n}}{1+a^{2n}}$; $a > 0$.

Find the limit $\lim_{n \to \infty } \frac{a^{n}-a^{-n}}{a^{n}+a^{-n}}$; $a > 0$.

Find the limit $\lim_{n \to \infty } \frac{a^{n}}{(1+a)(1+a^{2}) \cdots (1+a^{n})}$; $a > 0$.

Find the limit $\lim_{x \to 0} \frac{\sin 4x}{x}$.

Find the limit $\lim_{x \to 0} \frac{\sin 2x}{\sin 3x}$.

Find the limit $\lim_{x \to 0} \frac{\sin 5x}{\operatorname {tg} x}$.

Find the limit $\lim_{x \to 0} \frac{\sin mx}{\sin nx}$.

Find the limit $\lim_{x \to \pi } \frac{\sin mx}{\sin nx}$, where $m$ and $n$ are integers.

Find the limit $\lim_{x \to 0} x \operatorname {ctg} x$.

Find the limit $\lim_{n \to \infty } 2^{n} \sin \frac{x}{2^{n}}$.

Find the limit $\lim_{x \to 0} \frac{1-\cos x}{x^{2}}$.

Find the limit $\lim_{x \to 0} \frac{\cos mx-\cos nx}{x^{2}}$.

Find the limit $\lim_{x \to 1} (1-x) \operatorname {tg} \frac{\pi x}{2}$.

Find the limit $\lim_{x \to 0} \frac{\operatorname {tg} x-\sin x}{x^{3}}$.

Find the limit $\lim_{x \to 0} \frac{\sin (a+x)-\sin (a-x)}{x}$.

Find the limit $\lim_{x \to 0} \frac{\sin (a+x)+\sin (a-x)-2 \sin a}{x^{2}}$.

Find the limit $\lim_{x \to 0} \frac{\cos (a+x)+\cos (a-x)-2 \cos a}{1-\cos x}$.

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

Find the limit $\lim_{x \to 0} \frac{\sqrt{\cos x}-1}{x^{2}}$.

Find the limit $\lim_{x \to \pi /2} \frac{\cos \frac{x}{2}-\sin \frac{x}{2}}{\cos x}$.

Find the limit $\lim_{x \to \pi /3} \frac{\sin \left(x-\frac{\pi }{3}\right)}{1-2\cos x}$.

Find the limit $\lim_{x \to \pi /4} \frac{\sqrt{2}\cos x-1}{1-\operatorname {tg}^{2} x}$.

Find the limit $\lim_{x \to 0} \frac{\sqrt{1+\operatorname {tg} x}-\sqrt{1-\operatorname {tg} x}}{\sin x}$.

Find the limit $\lim_{x \to 0} \frac{\sqrt[m]{\cos \alpha x}-\sqrt[m]{\cos \beta x}}{x^{2}}$.

Find the limit $\lim_{x \to 0} \frac{\cos x-\sqrt[3]{\cos x}}{\sin^{2} x}$.

Find the limit $\lim_{x \to 0} \frac{1-\cos x \sqrt{\cos 2x}}{x^{2}}$.

Find the limit $\lim_{x \to 0} \frac{\sqrt{1+x \sin x}-\cos x}{\sin^{2} \frac{x}{2}}$.

Find the limit $\lim_{x \to \infty } \left(\frac{x+1}{x-1}\right)^{x}$.

Find the limit $\lim_{x \to \infty } \left(\frac{2x+3}{2x+1}\right)^{x+1}$.

Find the limit $\lim_{x \to \infty } \left(\frac{x^{2}-1}{x^{2}}\right)^{x^{4}}$.

Find the limit $\lim_{x \to 0} (1+\operatorname {tg} x)^{\operatorname {ctg} x}$.

Find the limit $\lim_{x \to 0} (1+3 \operatorname {tg}^{2} x)^{\operatorname {ctg}^{2} x}$.

Find the limit $\lim_{x \to 0} \left(\frac{\cos x}{\cos 2x}\right)^{1/x^{2}}$.

Find the limit $\lim_{m \to \infty } \left(\cos \frac{x}{m}\right)^{m}$.

Find the limit $\lim_{m \to \infty } \left(\cos \frac{x}{\sqrt{m}}\right)^{m}$.

Find the limit $\lim_{m \to \infty } \left(\cos \frac{x}{m}+\lambda \sin \frac{x}{m}\right)^{m}$.

Find the limit $\lim_{x \to a} \left(\frac{\sin x}{\sin a}\right)^{1/(x-a)}$.

Find the limit $\lim_{m \to \infty } \left(\frac{\sin \left(a+\frac{b}{m}\right)}{\sin a}\right)^{m}$.

Find the limit $\lim_{x \to \pi /4} \operatorname {tg} x^{\operatorname {tg} 2x}$.

Find the limit $\lim_{n \to \infty } \left(\sin \frac{n+1}{n-1} \frac{\pi }{2}\right)^{\operatorname {tg} \frac{n+1}{n-1} \frac{\pi }{2}}$.

Find the limit $\lim_{x \to 0} \left(\operatorname {tg}\left(\frac{\pi }{4}+x\right)\right)^{\operatorname {ctg} 2x}$.

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

Find the limit $\lim_{x \to \pi /4} \left(\operatorname {tg}\left(\frac{\pi }{8}+x\right)\right)^{\operatorname {tg} 2x}$.

Find the limit $\lim_{n \to \infty } \left(\sin \left(\frac{2n+1}{2n-1} \cdot \frac{\pi }{4}\right)\right)^{\operatorname {tg} \frac{n+1}{n+3} \cdot \frac{\pi }{2}}$; $n$ is a positive integer.

Find the limit $\lim_{x \to e} \frac{\ln x-1}{x-e}$.

Find the limit $\lim_{x \to 10} \frac{\lg x-1}{x-10}$.

Find the limit $\lim_{x \to \pi /4} \frac{\ln \operatorname {tg} x}{\cos 2x}$.

Find the limit $\lim_{n \to \infty } n^{2} \ln \cos \frac{\pi }{n}$.

Find the limit $\lim_{x \to 0} \frac{\ln \cos \alpha x}{\ln \cos \beta x}$.

Find the limit $\lim_{n \to \infty } n \ln \operatorname {tg}\left(\frac{\pi }{4}+\frac{\pi }{n}\right)$.

Find the limit $\lim_{x \to \pi /4} \frac{\ln \operatorname {tg} x}{1-\operatorname {ctg} x}$.

Find the limit $\lim_{x \to +\infty } \frac{\ln (e^{x}+1)}{x}$.

Find the limit $\lim_{x \to +\infty } \frac{\ln (1+e^{\alpha x})}{\ln (1+e^{\beta x})}$; $\alpha > 0$, $\beta > 0$.

Find the limit $\lim_{n \to \infty } \left(a^{1/n}-1\right) n$.

Find the limit $\lim_{n \to \infty } n^{2}\left(\sqrt[n]{a}-\sqrt[n+1]{a}\right)$; $a > 0$.

Find the limit $\lim_{n \to \infty } n^{2} \left(a^{1/n}+a^{-1/n}-2\right)$.

Find the limit $\lim_{x \to 0} \frac{a^{x}-b^{x}}{x}$; $a > 0$, $b > 0$.

Find the limit $\lim_{x \to 0} \frac{e^{\alpha x}-e^{\beta x}}{x}$.

Find the limit $\lim_{x \to 0} \frac{e^{\alpha x}-e^{\beta x}}{\sin \alpha x-\sin \beta x}$.

Find the limit $\lim_{x \to c} \frac{a^{x}-a^{c}}{x-c}$; $a > 0$.

Find the limit $\lim_{n \to \infty } (1+a^{n})^{1/n}$; $a > 0$.

Find the limit $\lim_{n \to \infty } \left(\frac{\sqrt[n]{a}+\sqrt[n]{b}}{2}\right)^{n}$; $a > 0$, $b > 0$.

Find the limit $\lim_{n \to \infty } \left(\frac{\sqrt[n]{a_{1}}+\sqrt[n]{a_{2}}+\ldots +\sqrt[n]{a_{m}}}{m}\right)^{n}$; all $a_{i} > 0$.

Find the limit $\lim_{x \to 0} \frac{\ln (1+x+x^{2})+\ln (1-x+x^{2})}{x^{2}}$.

Find the limit $\lim_{x \to \infty } x \ln \frac{2a+x}{a+x}$.