91Ó°ÊÓ

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

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0}\left(x \ln x^{1000}\right) $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{t \rightarrow 1} \frac{\sqrt{t}-t^{2}}{\ln t} $$

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

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} 3 x^{2} \csc ^{2} x $$

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

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{7^{\sqrt{x}}-1}{2^{\sqrt{x}}-1} $$

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

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

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\ln \cos 2 x}{7 x^{2}} $$