91Ó°ÊÓ

In Problems \(1-42,\) find the limits. $$ \lim _{x \rightarrow \infty} \frac{2 x+1}{\sqrt{x^{2}+3}} . \text { Hint: Divide numerator and denominator } $$ $$ \begin{aligned} &\text { by }\\\ &x \text { . Note that, for } x>0, \sqrt{x^{2}+3 / x}=\sqrt{\left(x^{2}+3\right) / x^{2}} \text { . } \end{aligned} $$

In Problems \(1-22,\) find the indicated limit or state that it does not exist. $$ \lim _{t \rightarrow 2} \frac{t+2}{(t-2)^{2}} $$

Find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$ \lim _{x \rightarrow 1} \frac{x^{2}+x-2}{x^{2}-1} $$

In Problems \(1-42,\) find the limits. $$ \lim _{x \rightarrow \infty} \frac{\sqrt{2 x+1}}{x+4} $$

In Problems \(1-22,\) find the indicated limit or state that it does not exist. $$ \lim _{x \rightarrow 0^{+}} \frac{\cos x}{x} $$

Give an \(\varepsilon-\delta\) proof of each limit fact. \(\lim _{x \rightarrow 1}\left(2 x^{2}+1\right)=3\)

Find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$ \lim _{x \rightarrow-3} \frac{x^{2}-14 x-51}{x^{2}-4 x-21} $$

In Problems \(19-28\), use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{t \rightarrow 0} \frac{1-\cos t}{2 t} $$

Find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$ \lim _{u \rightarrow-2} \frac{u^{2}-u x+2 u-2 x}{u^{2}-u-6} $$

Give an \(\varepsilon-\delta\) proof of each limit fact. \(\lim _{x \rightarrow-1}\left(x^{2}-2 x-1\right)=2\)