Chapter 2

Simplify the given expression. $$ e^{\ln x^{2}-y \ln x} $$

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

Find the limits. $$ \lim _{\theta \rightarrow \infty} \frac{\sin ^{2} \theta}{\theta^{2}-5} $$

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. $$ \lim _{x \rightarrow 0}(2 x-1)=-1 $$

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

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

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

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

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

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ f(t)=\left\\{\begin{array}{ll} t-3 & \text { if } t \leq 3 \\ 3-t & \text { if } t>3 \end{array}\right. $$

In Problems 13-24, 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. $$ \lim _{x \rightarrow 2} \frac{x^{2}-4}{x^{2}+4} $$

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}=10 $$

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

Make use of the known graph of \(y=\ln x\) to sketch the graphs of the equations. $$ y=\ln |x| $$

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

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ f(t)=\left\\{\begin{array}{cl} t^{2}-9 & \text { if } t \leq 3 \\ (3-t)^{2} & \text { if } t>3 \end{array}\right. $$

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

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

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

Make use of the known graph of \(y=\ln x\) to sketch the graphs of the equations. $$ y=\ln \sqrt{x} $$

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

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ f(x)=\left\\{\begin{array}{ll} -3 x+7 & \text { if } x \leq 3 \\ -2 & \text { if } x>3 \end{array}\right. $$

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

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

Make use of the known graph of \(y=\ln x\) to sketch the graphs of the equations. $$ y=\ln \left(\frac{1}{x}\right) $$

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 limits. $$ \lim _{n \rightarrow \infty} \frac{n}{2 n+1} $$

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

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

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

Make use of the known graph of \(y=\ln x\) to sketch the graphs of the equations. $$ y=\ln (x-2) $$

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

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.

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

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow 4} \frac{\sqrt{2 x-1}}{\sqrt{x-3}}=\sqrt{7} $$

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

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 limits. $$ \lim _{n \rightarrow \infty} \frac{n^{2}}{n+1} $$

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

Find each of the following limits. (a) \(\lim _{x \rightarrow 0}(1+x)^{1000}\) (b) \(\lim _{x \rightarrow 0}(1)^{1 / x}\) (c) \(\lim _{x \rightarrow 0^{+}}(1+\varepsilon)^{1 / x}, \varepsilon>0\) (d) \(\lim _{x \rightarrow 0^{-}}(1+\varepsilon)^{1 / x}, \varepsilon>0\)

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

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow 1} \frac{14 x^{2}-20 x+6}{x-1}=8 $$

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

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