91Ó°ÊÓ

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow-5} \frac{x^{2}-25}{x+5}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\ln \left(t^{3 / 2}\right)$$

Show that each function is a solution of the given initial value problem. $$\begin{array}{lll} \begin{array}{l} \text { Differential } \\ \text { equation } \end{array} & \begin{array}{l} \text { Initial } \\ \text { equation } \end{array} & \begin{array}{l} \text { Solution } \\ \text { candidate } \end{array} \\ \hline \begin{aligned} &x^{2} y^{\prime}=x y-y^{2}\\\ &x>1 \end{aligned} &y(e)=e&y=\frac{x}{\ln x} \end{array}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=x e^{x}-e^{x}$$

Solve the differential equations. $$2 \sqrt{x y} \frac{d y}{d x}=1, \quad x, y>0$$

Use l'Hôpital's rule to find the limits. $$\lim _{t \rightarrow-3} \frac{t^{3}-4 t+15}{t^{2}-t-12}$$

True, or false? As \(x \rightarrow \infty\) a. \(\quad x=o(x)\) b. \(x=o(x+5)\) c. \(x=O(x+5)\) d. \(x=O(2 x)\) e. \(e^{x}=o\left(e^{2 x}\right)\) f. \(x+\ln x=O(x)\) g. \(\ln x=o(\ln 2 x)\) h. \(\sqrt{x^{2}+5}=O(x)\)

Find the values. $$\sin \left(\cos ^{-1}\left(\frac{\sqrt{2}}{2}\right)\right)$$

Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$(\sinh x+\cosh x)^{4}$$

Find the values. $$\sec \left(\cos ^{-1} \frac{1}{2}\right)$$