Chapter 2

$$ \lim _{x \rightarrow-21}(3 x-1)=-64 $$

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

Evaluate each limit. $$ \lim _{t \rightarrow 0} \frac{\tan 2 t}{\sin 2 t-1} $$

Find the limits. \(\lim _{x \rightarrow \infty} \sqrt[3]{\frac{\pi x^{3}+3 x}{\sqrt{2 x^{3}+7 x}}}\)

In Problems 13-16, make use of the known graph of \(y=\ln x\) to sketch the graphs of the equations. \(y=\ln |x|\)

In Problems 7-18, find the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2). $$ \lim _{t \rightarrow 2} \frac{\sqrt{(t+4)(t-2)^{4}}}{(3 t-6)^{2}} $$

$$ \lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}=10 $$

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ f(t)= \begin{cases}t-3 & \text { if } t \leq 3 \\ 3-t & \text { if } t>3\end{cases} $$

Evaluate each limit. $$ \lim _{t \rightarrow 0} \frac{\sin 3 t+4 t}{t \sec t} $$

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. $$\lim _{x \rightarrow 2} \frac{x^{2}-4}{x^{2}+4}$$

Find the limits. \(\lim _{x \rightarrow \infty} \sqrt[3]{\frac{1+8 x^{2}}{x^{2}+4}}\)

In Problems 13-16, make use of the known graph of \(y=\ln x\) to sketch the graphs of the equations. \(y=\ln \sqrt{x}\)

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

$$ \lim _{x \rightarrow 0}\left(\frac{2 x^{2}-x}{x}\right)=-1 $$

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ f(t)=\left\\{\begin{array}{cc} t^{2}-9 & \text { if } t \leq 3 \\ (3-t)^{2} & \text { if } t>3 \end{array}\right. $$

Evaluate each limit. $$ \lim _{\theta \rightarrow 0} \frac{\sin ^{2} \theta}{\theta^{2}} $$

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. $$\lim _{x \rightarrow 2} \frac{x^{2}-5 x+6}{x-2}$$

Find the limits. \(\lim _{x \rightarrow \infty} \sqrt{\frac{x^{2}+x+3}{(x-1)(x+1)}}\)

In Problems 7-18, find the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2). $$ \lim _{x \rightarrow 3} \frac{x^{4}-18 x^{2}+81}{(x-3)^{2}} $$

$$ \lim _{x \rightarrow 5} \frac{2 x^{2}-11 x+5}{x-5}=9 $$

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ f(x)= \begin{cases}-3 x+7 & \text { if } x \leq 3 \\ -2 & \text { if } x>3\end{cases} $$

Plot the functions \(u(x), l(x)\), and \(f(x)\). Then use these graphs along with the Squeeze Theorem to determine \(\lim _{x \rightarrow 0} f(x)\). $$ u(x)=|x|, l(x)=-|x|, f(x)=x \sin (1 / x) $$

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. $$\lim _{x \rightarrow-1} \frac{x^{2}-2 x-3}{x+1}$$

Find the limits. \(\lim _{n \rightarrow \infty} \frac{n}{2 n+1}\)

In Problems 13-16, make use of the known graph of \(y=\ln x\) to sketch the graphs of the equations. \(y=\ln (x-2)\)

In Problems 7-18, find the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2). $$ \lim _{u \rightarrow 1} \frac{(3 u+4)(2 u-2)^{3}}{(u-1)^{2}} $$

$$ \lim _{x \rightarrow 1} \sqrt{2 x}=\sqrt{2} $$

Plot the functions \(u(x), l(x)\), and \(f(x)\). Then use these graphs along with the Squeeze Theorem to determine \(\lim _{x \rightarrow 0} f(x)\). $$ u(x)=|x|, I(x)=-|x|, f(x)=x \sin \left(1 / x^{2}\right) $$

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. $$\lim _{x \rightarrow-1} \frac{x^{2}+x}{x^{2}+1}$$

Find the limits. \(\lim _{n \rightarrow \infty} \frac{n^{2}}{n^{2}+1}\)

Sketch the graph of \(y=\ln \cos x+\ln \sec x\) on \((-\pi / 2, \pi / 2)\), but think before you begin.

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

$$ \lim _{x \rightarrow 4} \frac{\sqrt{2 x-1}}{\sqrt{x-3}}=\sqrt{7} $$

Plot the functions \(u(x), l(x)\), and \(f(x)\). Then use these graphs along with the Squeeze Theorem to determine \(\lim _{x \rightarrow 0} f(x)\). $$ u(x)=|x|, l(x)=-|x|, f(x)=\left(1-\cos ^{2} x\right) / x $$

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. $$\lim _{x \rightarrow-1} \frac{x^{3}-6 x^{2}+11 x-6}{x^{3}+4 x^{2}-19 x+14}$$

Find the limits. \(\lim _{n \rightarrow \infty} \frac{n^{2}}{n+1}\)

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

In Problems 18-23, the given function is not defined at a certain point. How should it be defined in order to make it continuous at that point? (See Example 1.) $$ f(x)=\frac{x^{2}-49}{x-7} $$

Plot the functions \(u(x), l(x)\), and \(f(x)\). Then use these graphs along with the Squeeze Theorem to determine \(\lim _{x \rightarrow 0} f(x)\). $$ u(x)=1, l(x)=1-x^{2}, f(x)=\cos ^{2} x $$

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. $$\lim _{x \rightarrow 2} \frac{x^{2}+7 x+10}{x+2}$$

Find the limits. \(\lim _{n \rightarrow \infty} \frac{n}{n^{2}+1}\)

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

Plot the functions \(u(x), l(x)\), and \(f(x)\). Then use these graphs along with the Squeeze Theorem to determine \(\lim _{x \rightarrow 0} f(x)\). $$ u(x)=2, l(x)=2-x^{2}, f(x)=1+\frac{\sin x}{x} $$

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. $$\lim _{x \rightarrow 1} \frac{x^{2}+x-2}{x^{2}-1}$$

Find the limits. \(\lim _{x \rightarrow \infty} \frac{2 x+1}{\sqrt{x^{2}+3}}\)

Find each of the following limits. (a) \(\lim _{n \rightarrow \infty}\left(1+\frac{2}{n}\right)^{100}\) limits. (b) \(\lim _{n \rightarrow \infty}(1.001)^{n}\) (c) \(\lim _{n \rightarrow \infty}\left(\frac{n+3}{n}\right)^{n-1}\)

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

$$ \lim _{x \rightarrow 1}\left(2 x^{2}+1\right)=3 $$

Prove that \(\lim _{t \rightarrow c} \cos t=\cos c\) using an argument similar to the one used in the proof that \(\lim _{t \rightarrow c} \sin t=\sin c\).

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. $$\lim _{x \rightarrow-3} \frac{x^{2}-14 x-51}{x^{2}-4 x-21}$$