L'Hôpital's Rule

Find the limit of the functions. $\lim_{x \rightarrow 1} \frac{3 x^{2}+4 x-7}{2 x^{2}+3 x-5}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{\ln \cos x}{\ln \cos 3 x}$.

Find the limit of the functions. $\lim_{x \rightarrow 3} \frac{\ln \left(x^{2}-8\right)}{2 x^{2}-5 x-3}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{\operatorname {ch} 2 x-1}{x^{2}}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{\sin a x-\sin b x}{\operatorname {sh} a x-\operatorname {sh} b x}, a \neq b$.

Find the limit of the functions. $\lim_{x \rightarrow 1} \frac{x^{\alpha }-1}{x^{\beta }-1}, \beta \neq 0$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{\ln \cos a x}{x^{2}}$.

Find the limit of the functions. $\lim_{x \rightarrow a} \frac{x^{a}-a^{x}}{a^{x}-a^{a}}, a > 0, a \neq 1$.

Find the limit of the functions. $\lim_{x \rightarrow a} \frac{x^{a}-a^{x}}{x^{a}-a^{a}}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{\ln (1+x)-x}{\operatorname {tg}^{2} x}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{\ln ((2 / \pi ) \arccos x)}{\ln (1+x)}$.

Find the limit of the functions. $\lim_{x \rightarrow \pi / 4} \frac{\ln \operatorname {tg} x}{\operatorname {ctg} 2 x}$.

Find the limit of the functions. $\lim_{x \rightarrow \pi / 6} \frac{4 \sin^{2} x-6 \sin x+1}{3 \sin^{2} x+5 \sin x-4}$.

Find the limit of the functions. $\lim_{x \rightarrow 1} \frac{x^{5}-3 x^{2}+7 x-5}{x^{4}-5 x+4}$.

Find the limit of the functions. $\lim_{x \rightarrow 1} \frac{x^{x}-1}{\ln x}$.

Find the limit of the functions. $\lim_{x \rightarrow 1} \frac{\sqrt{5 x^{3}-x}-2 x}{\sqrt[5]{x^{2}}-1}$.

Find the limit of the functions. $\lim_{x \rightarrow 1+0} \frac{\operatorname {arctg}(x-1)}{\sqrt{x^{2}+x-2}}$.

Find the limit of the functions. $\lim_{x \rightarrow 1} \frac{x^{3}-3 x^{2}+7 x-5}{x^{3}+2 x^{2}-9 x+6}$.

Find the limit of the functions. $\lim_{x \rightarrow \pi / 6} \frac{\sqrt[5]{3 \operatorname {tg}^{2} x}-1}{2 \sin^{2} x+5 \sin x-3}$.

Find the limit of the functions. $\lim_{x \rightarrow \pi / 2} \frac{\cos (2 m+1) x}{\cos (2 n+1) x}, m \in \mathbb {N}, n \in \mathbb {N}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{x \arcsin x^{2}}{x \cos x-\sin x}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{x-\sin x}{\operatorname {tg} x-x}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{(x+1) \ln (1+x)-x}{e^{x}-x-1}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{\ln ((1+x) /(1-x))-2 x}{x-\sin x}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{(a+x)^{x}-a^{x}}{x^{2}}, a > 0$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{2 \operatorname {tg} 3 x-6 \operatorname {tg} x}{3 \operatorname {arctg} x-\operatorname {arctg} 3 x}$.

Find the limit of the functions. $\lim_{x \rightarrow 1} \frac{x^{20}-2 x+1}{x^{30}-2 x+1}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{\operatorname {tg} x-x}{\ln^{3}(1+x)}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{\sin 2 x-2 x}{x^{2} \arcsin x}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{\operatorname {tg} x-x}{\arcsin x-\ln (1+x)}$.

Find the limit of the functions. $\lim_{x \rightarrow 1}\left[\frac{x^{10}-10 x+9}{(x-1)^{2}}\right]$.

Find the limit of the functions. $\lim_{x \rightarrow 1} \frac{x^{50}-50 x+49}{x^{100}-100 x+99}$.

Find the limit of the functions. $\lim_{x \rightarrow -1} \frac{2 x^{4}+3 x^{3}-4 x^{2}-9 x-4}{3 x^{4}+5 x^{3}+3 x^{2}+3 x+2}$.

Find the limit of the functions. $\lim_{x \rightarrow 1} \frac{\alpha x^{\alpha +2}-(\alpha +1) x^{\alpha +1}+x}{(x-1)^{2}}$.

Find the limit of the functions. $\lim_{x \rightarrow 1} \frac{\alpha \left(1-x^{\beta }\right)-\beta \left(1-x^{\alpha }\right)}{\left(1-x^{\alpha }\right)\left(1-x^{\beta }\right)}, \alpha \beta \neq 0$

Find the limit of the functions. $\lim_{x \rightarrow 1} \frac{\ln x-x+1}{x-x^{x}}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \frac{e^{\sin x}-e^{x}}{\sin x-x}$.

Find the limit of the functions. $\lim_{x \rightarrow -1} \frac{x^{4}+x^{3}-3 x^{2}-5 x-2}{x^{4}+2 x^{3}-2 x-1}$.

Find the limit of the functions. $\lim_{x \rightarrow +0} \frac{\ln x}{\ln \sin x}$.

Find the limit of the functions. $\lim_{x \rightarrow +0} \frac{\ln \sin x}{\operatorname {ctg} x}$.

Find the limit of the functions. $\lim_{x \rightarrow \pi / 2+0} \frac{\ln (x-\pi / 2)}{\operatorname {tg} x}$.

Find the limit of the functions. $\lim_{x \rightarrow +0} \frac{\ln (1-\cos x)}{\ln \operatorname {tg} x}$.

Find the limit of the functions. $\lim_{x \rightarrow +0} \frac{3+\ln x}{2-3 \ln \sin x}$.

Find the limit of the functions. $\lim_{x \rightarrow +\infty } \frac{x^{\alpha } \ln^{\beta } x}{e^{\gamma x}}$.

Find the limit of the functions. $\lim_{x \rightarrow +\infty } \frac{\sqrt[3]{x} \ln \ln x}{\sqrt[3]{2 x+3} \sqrt{\ln x}}$.

Find the limit of the functions. $\lim_{x \rightarrow +\infty } \frac{e^{x}}{x^{x^{\alpha } \ln^{\beta } x}}$.

Find the limit of the functions. $\lim_{x \rightarrow 0} \sin x \ln \operatorname {ctg} x$.

Find the limit of the functions. $\lim_{x \rightarrow +\infty } x \ln \left(\frac{2}{\pi } \operatorname {arctg} x\right)$.

Find the limit of the functions. $\lim_{x \rightarrow +\infty } x^{n} e^{-x^{3}}$.

Find the limit of the functions. $\lim_{x \rightarrow +\infty }(\pi -2 \operatorname {arctg} \sqrt{x}) \sqrt{x}$.

Find the limit of the functions. $\lim_{x \rightarrow +\infty } x\left(\pi -2 \arcsin \left(x / \sqrt{x^{2}+1}\right)\right)$.

Find the limit of the functions. $\lim_{x \rightarrow +0} x^{\alpha } \ln^{\beta }(1 / x), \alpha > 0, \beta > 0$.

Find the limit of the functions. $\lim_{x \rightarrow +0}\left(x^{x}-1\right) \ln x$.

Find the limit of the functions. $\lim_{x \rightarrow +\infty } x^{\alpha } a^{x}, a > 0, a \neq 1$.

Find the limit of the functions. $\lim_{x \rightarrow 0}\left(\frac{1}{\sin x}-\frac{1}{x}\right)$.

Find the limit of the functions. $\lim_{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\arcsin x}\right)$.

Find the limit of the functions. $\lim_{x \rightarrow 0}\left(\frac{1}{x^{2}}-\frac{1}{\sin^{2} x}\right)$.

Find the limit of the functions. $\lim_{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{e^{x}-1}\right)$.

Find the limit of the functions. $\lim_{x \rightarrow 0}\left(\frac{1}{x \operatorname {arctg} x}-\frac{1}{x^{2}}\right)$.

Find the limit of the functions. $\lim_{x \rightarrow +\infty }\left(x^{7 / 8}-x^{6 / 7} \ln^{2} x\right)$.

Find the limit of the functions. $\lim_{x \rightarrow 1}\left(\frac{\alpha }{1-x^{\alpha }}-\frac{\beta }{1-x^{\beta }}\right), \alpha \beta \neq 0$.

Find the limit of the functions. $\lim_{x \rightarrow 1} x^{1 /(x-1)}$.

Find the limit of the functions. $\lim_{x \rightarrow +\infty }\left(\frac{2}{\pi } \operatorname {arctg} x\right)^{x}$.

Find the limit of the functions. $\lim_{x \rightarrow 0}\left(\frac{2}{\pi } \arccos x\right)^{1 / x}$.

Find the limit of the functions. $\lim_{x \rightarrow 0}(\cos x)^{1 / x^{2}}$.

Find the limit of the functions. $\lim_{x \rightarrow +0}(1+x)^{\ln x}$.

Find the limit of the functions. $\lim_{x \rightarrow 0}\left(\frac{(1+x)^{1 / x}}{e}\right)^{1 / x}$.

Find the limit of the functions. $\lim_{x \rightarrow +0}(\arcsin x)^{\operatorname {tg} x}$.

Find the limit of the functions. $\lim_{x \rightarrow +0} x^{1 / \ln (\operatorname {sh} x)}$.

Find the limit of the functions. $\lim_{x \rightarrow \pi / 2-0}(\pi -2 x)^{\cos x}$.

Find the limit of the functions. $\lim_{x \rightarrow +0} x^{x^{x}-1}$.

Find the limit of the functions. $\lim_{x \rightarrow \pi / 2-0}(\operatorname {tg} x)^{\cos x}$.

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

Find the limit of the functions. $\lim_{x \rightarrow +0}|\ln x|^{2 x}$.

Find the limit of the functions. $\lim_{x \rightarrow +0}(1 / x)^{\sin x}$.

Show that the following limits cannot be computed using L'Hôpital's rule, and find these limits:

Determine whether L'Hôpital's rule can be applied to compute the limit

Find this limit, if it exists.

Find $\lim_{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}}$, assuming that $f^{\prime \prime }(a)$ exists.

Find $\lim_{h \rightarrow 0} \frac{f(a+3 h)-3 f(a+2 h)+3 f(a+h)-f(a)}{h^{3}}$, assuming that $f^{\prime \prime \prime }(a)$ exists.

Show that the function $f(x) = \begin{cases} e^{-1 / x^{2}}, & x \neq 0, \\ 0, & x = 0, \end{cases}$ is infinitely differentiable on the whole real line, and find $f^{(k)}(0), k \in \mathbb {N}$.

Find the limit of the function: