91Ó°ÊÓ

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{e^{x}-\ln (1+x)-1}{x^{2}} $$

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{\infty} \frac{d x}{\left(x^{2}+16\right)^{2}}\)

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

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\tan ^{-1} x-x}{8 x^{3}} $$

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0}\left(x+e^{x / 3}\right)^{3 / x} $$

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{\infty} \frac{1}{x^{2}+2 x+10} d x\)

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{\infty} \frac{x}{e^{2|x|}} d x\)

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

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\cosh x-1}{x^{2}} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{1-\cos x-x \sin x}{2-2 \cos x-\sin ^{2} x} $$