Chapter 12

Explain the difference between a value at \(x=a\) and the limit as \(x\) approaches \(a\).

Explain why we say a function does not have a limit as \(x\) approaches \(a\) if, as \(x\) approaches \(a,\) the left-hand limit is not equal to the right-hand limit.

For the following exercises, draw the graph of a function from the functional values and limits provided. $$\lim _{x \rightarrow 2^{-}} f(x)=2, \lim _{x \rightarrow 2^{+}} f(x)=-3, \lim _{x \rightarrow 0} f(x)=5, f(0)=1, f(1)=0$$

For the following exercises, draw the graph of a function from the functional values and limits provided. $$\lim _{x \rightarrow 3^{-}} f(x)=0, \lim _{x \rightarrow 3^{+}} f(x)=5, \lim _{x \rightarrow 5} f(x)=0, f(5)=4, f(3)\text { does not exist.}$$

For the following exercises, draw the graph of a function from the functional values and limits provided. $$\lim _{x \rightarrow 4} f(x)=6, \lim _{x \rightarrow 6^{+}} f(x)=-1, \lim _{x \rightarrow 0} f(x)=5, f(4)=6, f(2)=6$$

For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as \(x\) approaches 0. $$f(x)=(1+x)^{\frac{1}{x}}$$

For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as \(x\) approaches 0. $$g(x)=(1+x)^{\frac{2}{x}}$$

For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as \(x\) approaches 0. $$h(x)=(1+x)^{\frac{3}{x}}$$

For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as \(x\) approaches 0. $$i(x)=(1+x)^{\frac{4}{x}}$$

For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as \(x\) approaches 0. $$j(x)=(1+x)^{\frac{5}{x}}$$

For the following exercises, use numerical evidence to determine whether the limit exists at \(x=a\) . If not, describe the behavior of the graph of the function near \(x=a\) . Round answers to two decimal places. $$f(x)=\frac{x^{2}-4 x}{16-x^{2}} ; a=4$$

For the following exercises, use numerical evidence to determine whether the limit exists at \(x=a\) . If not, describe the behavior of the graph of the function near \(x=a\) . Round answers to two decimal places. $$f(x)=\frac{x^{2}-x-6}{x^{2}-9} ; a=3$$

For the following exercises, use numerical evidence to determine whether the limit exists at \(x=a\) . If not, describe the behavior of the graph of the function near \(x=a\) . Round answers to two decimal places. $$f(x)=\frac{x^{2}-6 x-7}{x^{2}-7 x} ; a=7$$

For the following exercises, use numerical evidence to determine whether the limit exists at \(x=a\) . If not, describe the behavior of the graph of the function near \(x=a\) . Round answers to two decimal places. $$f(x)=\frac{x^{2}-1}{x^{2}-3 x+2} ; a=1$$

For the following exercises, use numerical evidence to determine whether the limit exists at \(x=a\) . If not, describe the behavior of the graph of the function near \(x=a\) . Round answers to two decimal places. $$f(x)=\frac{10-10 x^{2}}{x^{2}-3 x+2} ; a=1$$

For the following exercises, use numerical evidence to determine whether the limit exists at \(x=a\) . If not, describe the behavior of the graph of the function near \(x=a\) . Round answers to two decimal places. $$f(x)=\frac{x}{6 x^{2}-5 x-6} ; a=\frac{3}{2}$$

For the following exercises, use numerical evidence to determine whether the limit exists at \(x=a\) . If not, describe the behavior of the graph of the function near \(x=a\) . Round answers to two decimal places. $$f(x)=\frac{x}{4 x^{2}+4 x+1} ; a=-\frac{1}{2}$$

For the following exercises, use numerical evidence to determine whether the limit exists at \(x=a\) . If not, describe the behavior of the graph of the function near \(x=a\) . Round answers to two decimal places. $$f(x)=\frac{2}{x-4} ; a=4$$

For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as \(x\) approaches the given value. $$\lim _{x \rightarrow 0} \frac{7 \tan x}{3 x}$$

For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as \(x\) approaches the given value. $$\lim _{x \rightarrow 4} \frac{x^{2}}{x-4}$$

For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as \(x\) approaches the given value. $$\lim _{x \rightarrow 0} \frac{2 \sin x}{4 \tan x}$$

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as \(x\) approaches \(a\). If the function has a limit as \(x\) approaches \(a\), state it. If not, discuss why there is no limit. $$\lim _{x \rightarrow 0} e^{-\frac{1}{x^{2}}}$$

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as \(x\) approaches \(a\). If the function has a limit as \(x\) approaches \(a\), state it. If not, discuss why there is no limit. $$\lim _{x \rightarrow 0} \frac{|x|}{x}$$

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as \(x\) approaches \(a\). If the function has a limit as \(x\) approaches \(a\), state it. If not, discuss why there is no limit. $$\lim _{x \rightarrow-1} \frac{|x+1|}{x+1}$$

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as \(x\) approaches \(a\). If the function has a limit as \(x\) approaches \(a\), state it. If not, discuss why there is no limit. $$\lim _{x \rightarrow-1} \frac{1}{(x+1)^{2}}$$

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as \(x\) approaches \(a\). If the function has a limit as \(x\) approaches \(a\), state it. If not, discuss why there is no limit. $$\lim _{x \rightarrow 1} \frac{1}{(x-1)^{3}}$$

Use numerical and graphical evidence to compare and contrast the limits of two functions whose formulas appear similar: \(f(x)=\left|\frac{1-x}{x}\right|\) and \(g(x)=\left|\frac{1+x}{x}\right|\) as \(x\) approaches. Use a graphing utility, if possible, to determine the left- and right-hand limits of the functions \(f(x)\) and \(g(x)\) as \(x\) approaches \(0 .\) If the functions have a limit as \(x\) approaches \(0,\) state it If not, discuss why there is no limit.

According to the Theory of Relativity, the mass \(m\) of a particle depends on its velocity \(v .\) That is $$m=\frac{m_{o}}{\sqrt{1-\left(v^{2} / c^{2}\right)}}$$ where \(m_{o}\) is the mass when the particle is at rest and \(c\) is the speed of light. Find the limit of the mass, \(m,\) as \(v\) approaches \(c^{-}\) .

52\. Allow the speed of light, \(c,\) to be equal to 1.0 . If the mass, \(m,\) is \(1,\) what occurs to \(m\) as \(v \rightarrow c ?\) Using the values listed in Table \(12.1,\) make a conjecture as to what the mass is as \(v\) approaches 1.00 . $$\begin{array}{|c|c|c|c|c|c|}\hline v & {0.5} & {0.9} & {0.95} & {0.99} & {0.999} & {0.99999} \\ \hline m & { 1.15} & {2.29} & {3.20} & {7.09} & {22.36} & {223.61} \\ \hline\end{array}$$

Give an example of a type of function \(f\) whose limit, as \(x\) approaches \(a,\) is \(f(a)\)

When direct substitution is used to evaluate the limit of a rational function as approaches \(a\) and the result is \(f(a)=\frac{0}{0},\) does this mean that the limit of \(f\) does not exist?

What does it mean to say the limit of \(f(x),\) as \(x\) approaches \(c,\) is undefined?

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow 0}(3) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow 2}\left(\frac{-5 x}{x^{2}-1}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow 2}\left(\frac{x^{2}-5 x+6}{x+2}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow 3}\left(\frac{x^{2}-9}{x-3}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow-1}\left(\frac{x^{2}-2 x-3}{x+1}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow \frac{3}{2}}\left(\frac{6 x^{2}-17 x+12}{2 x-3}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow-\frac{7}{2}}\left(\frac{8 x^{2}+18 x-35}{2 x+7}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow-3}\left(\frac{-7 x^{4}-21 x^{3}}{-12 x^{4}+108 x^{2}}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow 3}\left(\frac{x^{2}+2 x-3}{x-3}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{h \rightarrow 0}\left(\frac{(3+h)^{3}-27}{h}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{h \rightarrow 0}\left(\frac{(2-h)^{3}-8}{h}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{h \rightarrow 0}\left(\frac{(h+3)^{2}-9}{h}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{h \rightarrow 0}\left(\frac{\sqrt{5-h}-\sqrt{5}}{h}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow 0}\left(\frac{\sqrt{3-x}-\sqrt{3}}{x}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow 9}\left(\frac{x^{2}-81}{3-\sqrt{x}}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow 1}\left(\frac{\sqrt{x}-x^{2}}{1-\sqrt{x}}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow 0}\left(\frac{x}{\sqrt{1+2 x}-1}\right) $$

For the following exercises, evaluate the limits algebraically. $$ \lim _{x \rightarrow \frac{1}{2}}\left(\frac{x^{2}-\frac{1}{4}}{2 x-1}\right) $$