91Ó°ÊÓ

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow-\infty} x e^{x} $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d x}=\sec ^{2}\left(\frac{x}{2}\right),-1

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow 0^{+}}\left(\frac{1}{x}-\frac{1}{\sqrt{x}}\right) $$

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d x}=1+\sec ^{2}\left(\frac{x}{4}\right),-1

In Problems 59-72, solve the initial-value problem. $$ \frac{d y}{d x}=3 x^{2}, \text { for } x \geq 0 \text { with } y=1 \text { when } x=0 $$

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow 0^{+}} x^{3 x} $$

In Problems 59-72, solve the initial-value problem. $$ \frac{d y}{d x}=\frac{x^{2}}{3}, \text { for } x \geq 0 \text { with } y=2 \text { when } x=0 $$

Find the limits in Problems Be sure to check whether you can apply I'Hospital's rule before you evaluate the limit. $$ \lim _{x \rightarrow \infty}\left(\frac{x+1}{x+2}\right) $$

use l'Hospital's rule to find $$ \lim _{x \rightarrow 0} \frac{a^{x}-1}{b^{x}-1} $$ where \(a, b>0\).

In Problems 59-72, solve the initial-value problem. $$ \frac{d y}{d x}=2 \sqrt{x}, \text { for } x \geq 0 \text { with } y=2 \text { when } x=1 $$