91Ó°ÊÓ

Find the limits in Exercises \(13-20 .\) (if in doubt, look at the function's graph.) $$\lim _{x \rightarrow \infty} \csc ^{-1} x$$

Solve the differential equations in Exercises \(9-22\) . $$ y^{2} \frac{d y}{d x}=3 x^{2} y^{3}-6 x^{2} $$

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\operatorname{sech} \theta(1-\ln \operatorname{sech} \theta)$$

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\frac{1+\ln t}{t}$$

The function \(e^{x}\) outgrows any polynomial Show that \(e^{x}\) grows faster as \(x \rightarrow \infty\) than any polynomial $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$

Gives a formula for a function \(y=f(x)\) and shows the graphs of \(f\) and \(f^{-1} .\) Find a formula for \(f^{-1}\) in each case. $$f(x)=x^{2}, \quad x \leq 0$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) $$\lim _{x \rightarrow 1} \frac{x-1}{\ln x-\sin \pi x}$$

Solve the differential equations in Exercises \(9-22\) . $$ \frac{d y}{d x}=x y+3 x-2 y-6 $$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\ln \left(\frac{\sqrt{\theta}}{1+\sqrt{\theta}}\right)$$

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\operatorname{csch} \theta(1-\ln \operatorname{csch} \theta)$$