91Ó°ÊÓ

In Problems 7 - 18, find the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2). $$ \lim _{x \rightarrow 0} \frac{x^{4}+2 x^{3}-x^{2}}{x^{2}} $$

Evaluate each limit. $$ \lim _{t \rightarrow 0} \frac{\tan ^{2} 3 t}{2 t} $$

In Problems \(1-15,\) state whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ r(t)=\left\\{\begin{array}{ll} \frac{t^{3}-27}{t-3} & \text { if } t \neq 3 \\ 27 & \text { if } t=3 \end{array}\right. $$

Give an \(\varepsilon-\delta\) proof of each limit fact. \(\lim _{x \rightarrow 0}(2 x-1)=-1\)

In Problems \(1-42,\) find the limits. $$\lim _{x \rightarrow \infty} \frac{3 \sqrt{x^{3}}+3 x}{\sqrt{2 x^{3}}} $$

In Problems \(1-22,\) find the indicated limit or state that it does not exist. $$ \lim _{x \rightarrow 1 / 2^{+}}[4 x] $$

In Problems \(1-42,\) find the limits. $$ \lim _{x \rightarrow \infty} \sqrt[3]{\frac{\pi x^{3}+3 x}{\sqrt{2} x^{3}+7 x}} $$

In Problems 7 - 18, find the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2). $$ \lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3} $$

In Problems \(1-15,\) state whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ r(t)=\left\\{\begin{array}{ll} \frac{t^{3}-27}{t-3} & \text { if } t \neq 3 \\\ 23 & \text { if } t=3 \end{array}\right. $$

Give an \(\varepsilon-\delta\) proof of each limit fact. \(\lim _{x \rightarrow-21}(3 x-1)=-64\)