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

Evaluate the given improper integral or show that it diverges. $$ \int_{-1}^{1} \frac{d x}{1-x} $$

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

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'Hopital's Rule. \(\lim _{x \rightarrow 0} \frac{\int_{0}^{x} \sqrt{1+\sin t} d t}{x}\)

Evaluate the given improper integral or show that it diverges. $$ \int_{0}^{\infty} \frac{d x}{x+1} $$

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

Evaluate each improper integral or show that it diverges. $$ \int_{0}^{\infty} e^{-x} \cos x d x \text { Hint: Use table of integrals or a CAS } $$

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

Evaluate the given improper integral or show that it diverges. $$ \int_{1 / 2}^{2} \frac{d x}{x(\ln x)^{1 / 5}} $$