1.2E: Exercises

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Exercise $\PageIndex{1}$

In exercises 1 - 4, consider the graph of the function$y=f(x)$ shown here. Which of the statements about $y=f(x)$ are true and which are false? Explain why a statement is false.

$A graph of a piecewise function with three segments and a point. The first segment is a curve opening upward with vertex at (-8, -6). This vertex is an open circle, and there is a closed circle instead at (-8, -3). The segment ends at (-2,3), where there is a closed circle. The second segment stretches up asymptotically to infinity along x=-2, changes direction to increasing at about (0,1.25), increases until about (2.25, 3), and decreases until (6,2), where there is an open circle. The last segment starts at (6,5), increases slightly, and then decreases into quadrant four, crossing the x axis at (10,0). All of the changes in direction are smooth curves.$

1) $\displaystyle \lim_{x→10}f(x)=0$

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

Answer: False; $\displaystyle \lim_{x→−2^+}f(x)=+∞$

3) $\displaystyle \lim_{x→−8}f(x)=f(−8)$

4) $\displaystyle \lim_{x→6}f(x)=5$

Answer: False; $\displaystyle \lim_{x→6}f(x)$ DNE since $\displaystyle \lim_{x→6^−}f(x)=2$ and $\displaystyle \lim_{x→6^+}f(x)=5$.

Exercise $\PageIndex{2}$

In exercises 1 - 5, use the following graph of the function $y=f(x)$ to find the values, if possible. Estimate when necessary.

$A graph of a piecewise function with two segments. The first segment exists for x <=1, and the second segment exists for x > 1. The first segment is linear with a slope of 1 and goes through the origin. Its endpoint is a closed circle at (1,1). The second segment is also linear with a slope of -1. It begins with the open circle at (1,2).$

1) $\displaystyle \lim_{x→1^−}f(x)$

2) $\displaystyle \lim_{x→1^+}f(x)$

Answer: $2$

3) $\displaystyle \lim_{x→1}f(x)$

4) $\displaystyle \lim_{x→2}f(x)$

Answer: $1$

5) $f(1)$

Exercise $\PageIndex{3}$

In exercises 1 - 4, use the graph of the function $y=f(x)$ shown here to find the values, if possible. Estimate when necessary.

$A graph of a piecewise function with two segments. The first is a linear function for x < 0. There is an open circle at (0,1), and its slope is -1. The second segment is the right half of a parabola opening upward. Its vertex is a closed circle at (0, -4), and it goes through the point (2,0).$

1) $\displaystyle \lim_{x→0^−}f(x)$

Answer: $1$

2) $\displaystyle \lim_{x→0^+}f(x)$

3) $\displaystyle \lim_{x→0}f(x)$

Answer: DNE

4) $\displaystyle \lim_{x→2}f(x)$

Exercise $\PageIndex{4}$

In exercises 1 - 6, use the graph of the function $y=f(x)$ shown here to find the values, if possible. Estimate when necessary.

$A graph of a piecewise function with three segments, all linear. The first exists for x < -2, has a slope of 1, and ends at the open circle at (-2, 0). The second exists over the interval [-2, 2], has a slope of -1, goes through the origin, and has closed circles at its endpoints (-2, 2) and (2,-2). The third exists for x>2, has a slope of 1, and begins at the open circle (2,2).$

1) $\displaystyle \lim_{x→−2^−}f(x)$

Answer: $0$

2) $\displaystyle \lim_{x→−2^+}f(x)$

3) $\displaystyle \lim_{x→−2}f(x)$

Answer: DNE

4) $\displaystyle \lim_{x→2^−}f(x)$

5) $\displaystyle \lim_{x→2^+}f(x)$

Answer: $2$

6) $\displaystyle \lim_{x→2}f(x)$

Exercise $\PageIndex{5}$

1. In exercises a-c, use the graph of the function $y=g(x)$ shown here to find the values, if possible. Estimate when necessary.

$A graph of a piecewise function with two segments. The first exists for x>=0 and is the left half of an upward opening parabola with vertex at the closed circle (0,3). The second exists for x>0 and is the right half of a downward opening parabola with vertex at the open circle (0,0).$

a) $\displaystyle \lim_{x→0^−}g(x)$

Answer: $3$

b) $\displaystyle \lim_{x→0^+}g(x)$

c) $\displaystyle \lim_{x→0}g(x)$

Answer: DNE

2. In exercises a-c, use the graph of the function $y=h(x)$ shown here to find the values, if possible. Estimate when necessary.

$A graph of a function with two curves approaching 0 from quadrant 1 and quadrant 3. The curve in quadrant one appears to be the top half of a parabola opening to the right of the y axis along the x axis with vertex at the origin. The curve in quadrant three appears to be the left half of a parabola opening downward with vertex at the origin.$

a) $\displaystyle \lim_{x→0^−}h(x)$

b) $\displaystyle \lim_{x→0^+}h(x)$

Answer: $0$

c) $\displaystyle \lim_{x→0}h(x)$

Exercise $\PageIndex{6}$

In exercises a-e, use the graph of the function $y=f(x)$ shown here to find the values, if possible. Estimate when necessary.

$A graph with a curve and a point. The point is a closed circle at (0,-2). The curve is part of an upward opening parabola with vertex at (1,-1). It exists for x > 0, and there is a closed circle at the origin.$

a) $\displaystyle \lim_{x→0^−}f(x)$

Answer: $-2$

b) $\displaystyle \lim_{x→0^+}f(x)$

c) $\displaystyle \lim_{x→0}f(x)$

Answer: DNE

d) $\displaystyle \lim_{x→1}f(x)$

e) $\displaystyle \lim_{x→2}f(x)$

Answer: $0$

Exercise $\PageIndex{7}$

In exercises 1 - 5, sketch the graph of a function with the given properties.

1) $\displaystyle \lim_{x→2}f(x)=1, \quad \lim_{x→4^−}f(x)=3, \quad \lim_{x→4^+}f(x)=6, \quad x=4$ is not defined.

2) $\displaystyle \lim_{x→−∞}f(x)=0, \quad \lim_{x→−1^−}f(x)=−∞, \quad \lim_{x→−1^+}f(x)=∞,\quad \lim_{x→0}f(x)=f(0), \quad f(0)=1, \quad \lim_{x→∞}f(x)=−∞$

Answer

Answers may vary

$A graph of a piecewise function with two segments. The first segment is in quadrant three and asymptotically goes to negative infinity along the y axis and 0 along the x axis. The second segment consists of two curves. The first appears to be the left half of an upward opening parabola with vertex at (0,1). The second appears to be the right half of a downward opening parabola with vertex at (0,1) as well.$

3) $\displaystyle \lim_{x→−∞}f(x)=2, \quad \lim_{x→3^−}f(x)=−∞, \quad \lim_{x→3^+}f(x)=∞, \quad \lim_{x→∞}f(x)=2, \quad f(0)=-\frac{1}{3}$

4) $\displaystyle \lim_{x→−∞}f(x)=2,\quad \lim_{x→−2}f(x)=−∞,\quad \lim_{x→∞}f(x)=2,\quad f(0)=0$

Answer

Answer may vary

$A graph containing two curves. The first goes to 2 asymptotically along y=2 and to negative infinity along x = -2. The second goes to negative infinity along x=-2 and to 2 along y=2.$

5) $\displaystyle \lim_{x→−∞}f(x)=0,\quad \lim_{x→−1^−}f(x)=∞,\quad \lim_{x→−1^+}f(x)=−∞, \quad f(0)=−1, \quad \lim_{x→1^−}f(x)=−∞, \quad \lim_{x→1^+}f(x)=∞, \quad \lim_{x→∞}f(x)=0$

Exercise $\PageIndex{8}$

1) Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, $x$, is shown here. We are mainly interested in the location of the front of the shock, labeled $X_{SF}$ in the diagram.

$A graph in quadrant one of the density of a shockwave with three labeled points: p1 and p2 on the y axis, with p1 > p2, and xsf on the x axis. It consists of y= p1 from 0 to xsf, x = xsf from y= p1 to y=p2, and y=p2 for values greater than or equal to xsf.$

a. Evaluate $\displaystyle \lim_{x→X_{SF}^+}ρ(x)$.

b. Evaluate $\displaystyle \lim_{x→X_{SF}^−}ρ(x)$.

c. Evaluate $\displaystyle \lim_{x→X_{SF}}ρ(x)$. Explain the physical meanings behind your answers.

Answer: a. $ρ_2$ b. $ρ_1$ c. DNE unless $ρ_1=ρ_2$. As you approach $X_{SF}$ from the right, you are in the high-density area of the shock. When you approach from the left, you have not experienced the “shock” yet and are at a lower density.

2) A track coach uses a camera with a fast shutter to estimate the position of a runner with respect to time. A table of the values of position of the athlete versus time is given here, where $x$ is the position in meters of the runner and $t$ is time in seconds. What is $\displaystyle \lim_{t→2}x(t)$? What does it mean physically?

$t(sec)$	$x(m)$
1.75	4.5
1.95	6.1
1.99	6.42
2.01	6.58
2.05	6.9
2.25	8.5

Exercise $\PageIndex{9}$ Terms and Concepts

What are the ways in which a limit may fail to exist?
T/F: If $\lim\limits_{x\to1-}f(x)=5$, then $\lim\limits_{x\to1}f(x)=5$
T/F: If $\lim\limits_{x\to1-}f(x)=5$, then $\lim\limits_{x\to1+}f(x)=5$
T/F: If $\lim\limits_{x\to1}f(x)=5$, then $\lim\limits_{x\to1-}f(x)=5$
T/F: If $\lim\limits_{x\to 1^-}f(x)=-\infty$, then $\lim\limits_{x\to 1^+}f(x)=\infty$.
T/F: If $\lim\limits_{x\to 5}f(x)=\infty$, then $f$ has a vertical asymptote at $x=5$.

Answer: Under construction.

Exercise $\PageIndex{10}$: Finding limits using Graphs

Evaluate each expression using the given graph of $f(x)$.

$clipboard_e4f5a8bce92d31a43cde475d21c7bbb8f.png$

Answer: (a)2,(b)2,(c) 2,(d) 1, (e)DNE, (f)1

$146.PNG$

Answer: Under construction.

$147.PNG$

Answer: Under construction.

$148.PNG$

Answer

Under construction.

$1410.PNG$

Answer

Under construction.

5..

$1410.PNG$

Answer: Under construction.

$1411.PNG$

Answer: Under construction.

$1412.PNG$

Answer: Under construction.

Exercise $\PageIndex{11}$: Infinite limits

In Exercises 1-6, evaluate the given limits using the graph of the function.

1. $f(x) = \frac{1}{(x+1)^2}$

$169.PNG$

(a) $\lim\limits_{x\to -1^-}f(x)$

(b) $\lim\limits_{x\to -1^+}f(x)$

2. $f(x) = \frac{1}{(x-3)(x-5)^2}$

$1610.PNG$

(a) $\lim\limits_{x\to 3^-}f(x)$

(b) $\lim\limits_{x\to 3^+}f(x)$

(d) $\lim\limits_{x\to 5^-}f(x)$

(e) $\lim\limits_{x\to 5^+}f(x)$

(f) $\lim\limits_{x\to 5}f(x)$

3. $f(x) = \frac{1}{e^x+1}$

$1611.PNG$

(a) $\lim\limits_{x\to -\infty}f(x)$

(b) $\lim\limits_{x\to \infty}f(x)$

(d) $\lim\limits_{x\to 0^+}f(x)$

4. $f(x) = x^2\sin (\pi x)$

$1612.PNG$

(a) $\lim\limits_{x\to -\infty}f(x)$

(b) $\lim\limits_{x\to \infty}f(x)$

5. $f(x)=\cos (x)$

$1613.PNG$

(a) $\lim\limits_{x\to -\infty}f(x)$

(b) $\lim\limits_{x\to \infty}f(x)$

6. $f(x) = 2^x +10$

$1614.PNG$

(a) $\lim\limits_{x\to -\infty}f(x)$

(b) $\lim\limits_{x\to \infty}f(x)$

Answer: Under construction.

Exercise $\PageIndex{11}$: One sided limits

Given $ \displaystyle f(x)= \left\{ \begin{array}{ccc}
x^3+1 & \mbox{ if } x <0 \\
0 & \mbox{ if } x =0 \\
\sqrt{x+1}-2 & \mbox{ if } x >0 \\
\end{array}
\right.$

Identify each limit:

$ \displaystyle \lim_{x \to 0^-} f(x)$
$ \displaystyle \lim_{x \to 0^+} f(x)$
$ \displaystyle \lim_{x \to 0} f(x)$
$ \displaystyle \lim_{x \to -1} f(x)$
$ \displaystyle \lim_{x \to 3} f(x)$

Hint:: Draw a graph.

Answer: 11, -1, DNE, 0, 0

Solution:

$ \displaystyle \lim_{x \to 0^-} f(x)=0^3+11$
$ \displaystyle \lim_{x \to 0^+} f(x)= \sqrt{0+1}-2=-1 $
Since $ \displaystyle \lim_{x \to 0^-} f(x) \ne \displaystyle \lim_{x \to 0^+} f(x)$, $ \displaystyle \lim_{x \to 0} f(x) =$ DNE. \
$\displaystyle \lim_{x \to -1} f(x)=(-1)^3+1=0$
$ \displaystyle \lim_{x \to 3} f(x)= \sqrt{3+1}-2=0$

Exercise $\PageIndex{12}$: Infinite limits

$\displaystyle\lim_{y \to 6^-} \frac{y+6}{y^2-36}$

Answer

Since $\displaystyle\lim_{y \to 6^-} \frac{y+6}{y^2-36}= \frac{12}{0}$, and $ \frac{5.9+6}{(5.9)^2-36}<0,$

$\displaystyle\lim_{y \to 6^-} \frac{y+6}{y^2-36} =-\infty$.

Contributors and Attributions

Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. http://www.apexcalculus.com/

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

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