91Ó°ÊÓ

Use \(I^{\prime}\) Hôpital's Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2. $$\lim _{x \rightarrow \infty} \frac{2 x^{2}+3 x}{x^{3}+x+1}$$

Which of the following functions grow faster than \(\ln x\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(\ln x ?\) Which grow slower? a. \(\log _{2}\left(x^{2}\right)\) b. \(\log _{10} 10 x\) c. \(1 / \sqrt{x}\) d. \(1 / x^{2}\) e. \(x-2 \ln x\) f. \(e^{-x}\) g. \(\ln (\ln x)\) h. \(\ln (2 x+5)\)

Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$\sinh (2 \ln x)$$

Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$\cosh 5 x+\sinh 5 x$$

Use l'Hôpital's rule to find the limits. $$\lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=e^{5-7 x}$$

Order the following functions from slowest growing to fastest growing as \(x \rightarrow \infty\) a. \(e^{x}\) b. \(x^{x}\) c. \((\ln x)^{x}\) d. \(e^{x / 2}\)

Show that each function is a solution of the given initial value problem. $$\begin{array}{lll} \begin{array}{l} \text { Differential } \\ \text { equation } \end{array} & \begin{array}{l} \text { Initial } \\ \text { equation } \end{array} & \begin{array}{l} \text { Solution } \\ \text { candidate } \end{array} \\ \hline \begin{aligned} &x y^{\prime}+y=-\sin x\\\ &x>0 \end{aligned} &y\left(\frac{\pi}{2}\right)=0&y=\frac{\cos x}{x} \end{array}$$

Order the following functions from slowest growing to fastest growing as \(x \rightarrow \infty\) a. \(2^{x}\) b. \(x^{2}\) c. \((\ln 2)^{x}\) d. \(e^{x}\)

Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$\cosh 3 x-\sinh 3 x$$