$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 1.3E: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Exercise $$\PageIndex{1}$$

In exercises 1 - 4, use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s).

1) $$\displaystyle \lim_{x→0}\,(4x^2−2x+3)$$

Use constant multiple law and difference law:

$$\displaystyle \lim_{x→0}\,(4x^2−2x+3)=4\lim_{x→0}x^2−2\lim_{x→0}x+\lim_{x→0}3=0 + 0 + 3=3$$

2) $$\displaystyle \lim_{x→1}\frac{x^3+3x^2+5}{4−7x}$$

3) $$\displaystyle \lim_{x→−2}\sqrt{x^2−6x+3}$$

Use root law: $$\displaystyle \lim_{x→−2}\sqrt{x^2−6x+3}=\sqrt{\lim_{x→−2}(x^2−6x+3)}=\sqrt{19}$$

4) $$\displaystyle \lim_{x→−1}(9x+1)^2$$

Exercise $$\PageIndex{2}$$

In exercises 1 - 4, use direct substitution to evaluate the limit of each continuous function.

1) $$\displaystyle \lim_{x→7}x^2$$

$$\displaystyle \lim_{x→7}x^2\;=\;49$$

2) $$\displaystyle \lim_{x→−2}(4x^2−1)$$

3) $$\displaystyle \lim_{x→0}\frac{1}{1+\sin x}$$

$$\displaystyle \lim_{x→0}\frac{1}{1+\sin x}\;=\;1$$

4) $$\displaystyle \lim_{x→1}\frac{2−7x}{x+6}$$

$$\displaystyle \lim_{x→1}\frac{2−7x}{x+6}\;=\;−\frac{5}{7}$$

Exercise $$\PageIndex{3}$$

In exercises 1 - 9, use direct substitution to show that each limit leads to the indeterminate form $$0/0$$. Then, evaluate the limit analytically.

1) $$\displaystyle \lim_{x→4}\frac{x^2−16}{x−4}$$

$$\displaystyle \lim_{x→4}\frac{x^2−16}{x−4}=\frac{16−16}{4−4}=\frac{0}{0};$$
then, $$\displaystyle \lim_{x→4}\frac{x^2−16}{x−4}= \lim_{x→4}\frac{(x+4)(x−4)}{x−4}=\lim_{x→4}(x+4) = 4+4 =8$$

2) $$\displaystyle \lim_{x→2}\frac{x−2}{x^2−2x}$$

3) $$\displaystyle \lim_{x→6}\frac{3x−18}{2x−12}$$

$$\displaystyle \lim_{x→6}\frac{3x−18}{2x−12}=\frac{18−18}{12−12}=\frac{0}{0};$$
then, $$\displaystyle \lim_{x→6}\frac{3x−18}{2x− 12}=\lim_{x→6}\frac{3(x−6)}{2(x−6)}=\lim_{x→6}\frac{3}{2}=\frac{3}{2}$$

4) $$\displaystyle \lim_{h→0}\frac{(1+h)^2−1}{h}$$

5) $$\displaystyle \lim_{t→9}\frac{t−9}{\sqrt{t}−3}$$

$$\displaystyle \lim_{x→9}\frac{t−9}{\sqrt{t}−3}=\frac{9−9}{3−3}=\frac{0}{0};$$
then, $$\displaystyle \lim_{t→9}\frac{t−9}{\sqrt{t}−3} =\lim_{t→9}\frac{t−9}{\sqrt{t}−3}\frac{\sqrt{t}+3}{\sqrt{t}+3}=\lim_{t→9}\frac{(t−9)(\sqrt{t}+3)}{t - 9}=\lim_{t→9}(\sqrt{t}+3)=\sqrt{9}+3=6$$

6) $$\displaystyle \lim_{h→0}\frac{\dfrac{1}{a+h}−\dfrac{1}{a}}{h}$$, where $$a$$ is a real-valued constant

7) $$\displaystyle \lim_{x→1}\frac{x^3−1}{x^2−1}$$

8) $$\displaystyle \lim_{x→1/2}\frac{2x^2+3x−2}{2x−1}$$

$$\displaystyle \lim_{x→1/2}\frac{2x^2+3x−2}{2x−1}=\frac{\frac{1}{2}+\frac{3}{2}−2}{1−1}=\frac{0}{0};$$
then, $$\displaystyle \lim_{x→ 1/2}\frac{2x^2+3x−2}{2x−1}=\lim_{x→1/2}\frac{(2x−1)(x+2)}{2x−1}=\lim_{x→1/2}(x+2)=\frac{1}{2}+2=\frac{5}{2}$$

9) $$\displaystyle \lim_{x→−3}\frac{\sqrt{x+4}−1}{x+3}$$

Exercise $$\PageIndex{4}$$

In exercises 1 - 4, use direct substitution to obtain an undefined expression. Then, use the method used in Example 9 of this section to simplify the function and determine the limit.

1) $$\displaystyle \lim_{x→−2^−}\frac{2x^2+7x−4}{x^2+x−2}$$

$$−∞$$

2) $$\displaystyle \lim_{x→−2^+}\frac{2x^2+7x−4}{x^2+x−2}$$

3) $$\displaystyle \lim_{x→1^−}\frac{2x^2+7x−4}{x^2+x−2}$$

$$−∞$$

4) $$\displaystyle \lim_{x→1^+}\frac{2x^2+7x−4}{x^2+x−2}$$

Exercise $$\PageIndex{5}$$

In exercises 1 - 8, assume that $$\displaystyle \lim_{x→6}f(x)=4,\quad \lim_{x→6}g(x)=9$$, and $$\displaystyle \lim_{x→6}h(x)=6$$. Use these three facts and the limit laws to evaluate each limit.

1) $$\displaystyle \lim_{x→6}2f(x)g(x)$$

$$\displaystyle \lim_{x→6}2f(x)g(x)=2\left(\lim_{x→6}f(x)\right)\left(\lim_{x→6}g(x)\right)=2 (4)(9)=72$$

2) $$\displaystyle \lim_{x→6}\frac{g(x)−1}{f(x)}$$

3) $$\displaystyle \lim_{x→6}\left(f(x)+\frac{1}{3}g(x)\right)$$

$$\displaystyle \lim_{x→6}\left(f(x)+\frac{1}{3}g(x)\right)=\lim_{x→6}f(x)+\frac{1}{3}\lim_{x→6}g(x)=4+\frac{1}{3}(9)=7$$

4) $$\displaystyle \lim_{x→6}\frac{\big(h(x)\big)^3}{2}$$

5) $$\displaystyle \lim_{x→6}\sqrt{g(x)−f(x)}$$

$$\displaystyle \lim_{x→6}\sqrt{g(x)−f(x)}=\sqrt{\lim_{x→6}g(x)−\lim_{x→6}f(x)}=\sqrt{9-4}=\sqrt{5}$$

6) $$\displaystyle \lim_{x→6}x⋅h(x)$$

7) $$\displaystyle \lim_{x→6}[(x+1)⋅f(x)]$$

$$\displaystyle \lim_{x→6}[(x+1)f(x)]=\left(\lim_{x→6}(x+1)\right)\left(\lim_{x→6}f(x)\right)=7(4)=28$$

8) $$\displaystyle \lim_{x→6}(f(x)⋅g(x)−h(x))$$

Exercise $$\PageIndex{6}$$

[T] In exercises 1 - 3, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits.

1) $$f(x)=\begin{cases}x^2, & x≤3\\ x+4, & x>3\end{cases}$$

a. $$\displaystyle \lim_{x→3^−}f(x)$$

b. $$\displaystyle \lim_{x→3^+}f(x)$$

a. $$9$$; b.$$7$$

1) $$g(x)=\begin{cases}x^3−1, & x≤0\\1, & x>0\end{cases}$$

a. $$\displaystyle \lim_{x→0^−}g(x)$$

b. $$\displaystyle \lim_{x→0^+}g(x)$$

3) $$h(x)=\begin{cases}x^2−2x+1, & x<2\\3−x, & x≥2\end{cases}$$

a. $$\displaystyle \lim_{x→2^−}h(x)$$

b. $$\displaystyle \lim_{x→2^+}h(x)$$

Exercise $$\PageIndex{7}$$

In exercises 1 - 8, use the following graphs and the limit laws to evaluate each limit.

1) $$\displaystyle \lim_{x→−3^+}(f(x)+g(x))$$

2) $$\displaystyle \lim_{x→−3^−}(f(x)−3g(x))$$

$$\displaystyle \lim_{x→−3^−}(f(x)−3g(x))=\lim_{x→−3^−}f(x)−3\lim_{x→−3^−}g(x)=0+6=6$$

3) $$\displaystyle \lim_{x→0}\frac{f(x)g(x)}{3}$$

4) $$\displaystyle \lim_{x→−5}\frac{2+g(x)}{f(x)}$$

$$\displaystyle \lim_{x→−5}\frac{2+g(x)}{f(x)}=\frac{2+\left(\displaystyle \lim_{x→−5}g(x)\right)}{\displaystyle \lim_{x→−5}f(x)}=\frac{2+0}{2}=1$$

5) $$\displaystyle \lim_{x→1}(f(x))^2$$

6) $$\displaystyle \lim_{x→1}\sqrt{f(x)−g(x)}$$

$$\displaystyle \lim_{x→1}\sqrt[3]{f(x)−g(x)}=\sqrt[3]{\lim_{x→1}f(x)−\lim_{x→1}g(x)}=\sqrt[3]{2+5}=\sqrt[3]{7}$$

7) $$\displaystyle \lim_{x→−7}(x⋅g(x))$$

8) $$\displaystyle \lim_{x→−9}[x⋅f(x)+2⋅g(x)]$$

$$\displaystyle \lim_{x→−9}(xf(x)+2g(x))=\left(\lim_{x→−9}x\right)\left(\lim_{x→−9}f(x)\right)+2\lim_{x→−9}g(x)=(−9)(6)+2(4)=−46$$

Exercise $$\PageIndex{8}$$

1) [T] In physics, the magnitude of an electric field generated by a point charge at a distance $$r$$ in a vacuum is governed by Coulomb’s law: $$E(r)=\dfrac{q}{4πε_0r^2}$$, where $$E$$ represents the magnitude of the electric field, $$q$$ is the charge of the particle, $$r$$ is the distance between the particle and where the strength of the field is measured, and $$\dfrac{1}{4πε_0}$$ is Coulomb’s constant: $$8.988×109N⋅m^2/C^2$$.

a. Use a graphing calculator to graph $$E(r)$$ given that the charge of the particle is $$q=10^{−10}$$.

b. Evaluate $$\displaystyle \lim_{r→0^+}E(r)$$. What is the physical meaning of this quantity? Is it physically relevant? Why are you evaluating from the right?

a.

b. ∞. The magnitude of the electric field as you approach the particle q becomes infinite. It does not make physical sense to evaluate the negative distance.

2) [T] The density of an object is given by its mass divided by its volume: $$ρ=m/V.$$

a. Use a calculator to plot the volume as a function of density $$(V=m/ρ)$$, assuming you are examining something of mass $$8$$ kg ($$m=8$$).

b. Evaluate $$\displaystyle \lim_{x→0^+}V(\rho)$$ and explain the physical meaning.

Exercise $$\PageIndex{9}$$

Evaluate the following:

1. $$\displaystyle \lim\limits_{x\to3}x^2-3x+7$$
2. $$\displaystyle \lim\limits_{x\to\pi}\left ( \frac{x-3}{x+5}\right )^7$$
3. $$\displaystyle \lim\limits_{x\to3}4^{{x^3}-8x}$$
4. $$\displaystyle \lim\limits_{x\to0}\ln (1+x)$$
5. $$\displaystyle \lim\limits_{x\to\pi}\frac{x^2+3x+5}{5x^2-2x-3}$$
6. $$\displaystyle \lim\limits_{x\to\pi}\frac{3x+1}{1-x}$$
7. $$\displaystyle \lim\limits_{x\to6}\frac{x^2-4x-12}{x^2-13x+42}$$
8. $$\displaystyle \lim\limits_{x\to0}\frac{x^2+2x}{x^2-2x}$$
9. $$\displaystyle \lim\limits_{x\to2}\frac{x^2+6x-16}{x^2-3x+2}$$
10. $$\displaystyle \lim\limits_{x\to2}\frac{x^2-5x-14}{x^2+10x+16}$$
11. $$\displaystyle \lim\limits_{x\to-2}\frac{x^2-5x-14}{x^2+10x+16}$$
12. $$\displaystyle \lim\limits_{x\to-1}\frac{x^2+9x+8}{x^2-6x-7}$$

Under Construction

Exercise $$\PageIndex{10}$$

$$\displaystyle \lim_{x \to 0} \frac{\sqrt{x+4}-2}{x}$$

$$\displaystyle \lim_{x \to 0} \frac{\sqrt{x+4}-2}{x} = \frac{\sqrt{0+4}-2}{0} =\left[\frac{0}{0}\right]$$

= $$\displaystyle \lim_{x \to 0} \frac{(\sqrt{x+4}-2) (\sqrt{x+4}+2)}{x (\sqrt{x+4}+2)}$$

= $$\displaystyle \lim_{x \to 0} \frac{((x+4)-4) }{x (\sqrt{x+4}+2)}$$

= $$\displaystyle \lim_{x \to 0} \frac{x }{x (\sqrt{x+4}+2)}$$

= $$\displaystyle \lim_{x \to 0} \frac{1 }{(\sqrt{x+4}+2)}= \frac{1 }{(\sqrt{0+4}+2)}= \frac{1 }{4}$$.

Exercise $$\PageIndex{11}$$

Evaluate the following limits:

1. $$\displaystyle \lim\limits_{x\to4}\frac{1}{|4-x|}$$

$$\displaystyle \infty$$

2. $$\displaystyle \lim\limits_{x\to-1^-}\sqrt{1-x^2}$$

0

3. $$\displaystyle \lim\limits_{x\to-1^+}\sqrt{1-x^2}$$

0

4. $$\displaystyle \lim\limits_{x\to2}\frac{|x-2|}{x^2+x-6}$$

$$\displaystyle \mbox{dne}$$

5. $$\displaystyle \lim\limits_{x\to2}\frac{\frac{1}{x}-\frac{1}{2}}{x-2}$$

$$\displaystyle \frac{1}{4}$$

6. $$\displaystyle f(x)=\begin{cases} x^2 & \mbox{if } x \leq 1 \\ 2x & \mbox{if } x > 1\end{cases}$$