91Ó°ÊÓ

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

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. $$ \lim _{u \rightarrow-2} \frac{u^{2}-u x+2 u-2 x}{u^{2}-u-6} $$

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

The given function is not defined at a certain point. How should it be defined in order to make it continuous at that point? $$ H(t)=\frac{\sqrt{t}-1}{t-1} $$

Find the limits. $$ \begin{array}{l} \text { } \lim _{x \rightarrow \infty}\left(\sqrt{2 x^{2}+3}-\sqrt{2 x^{2}-5}\right) . \text { Hint: } \quad \text { Multiply and }\\\ \text { divide by } \sqrt{2 x^{2}+3}+\sqrt{2 x^{2}-5} \end{array} $$

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow 0} \frac{(x-\sin x)^{2}}{x^{2}} $$

Find the limits. $$ \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+2 x}-x\right) $$

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow 0} x^{4}=0 $$

The given function is not defined at a certain point. How should it be defined in order to make it continuous at that point? $$ \phi(x)=\frac{x^{4}+2 x^{2}-3}{x+1} $$