91Ó°ÊÓ

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{\infty} \operatorname{sech} x d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \pi / 2}(\sin x)^{\cos x} $$

Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \operatorname{csch} x d x\)

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{-}} \frac{\sin x+\tan x}{e^{x}+e^{-x}-2} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\int_{0}^{x} \sqrt{1+\sin t} d t}{x} $$

Evaluate each improper integral or show that it diverges. \(\int_{0}^{\infty} e^{-x} \cos x d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} x^{1 / x} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{t} \cos t d t}{x^{2}} $$

Evaluate each improper integral or show that it diverges. \(\int_{0}^{\infty} e^{-x} \sin x d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0}(\cos x)^{1 / x^{2}} $$