Skip to main content
Mathematics LibreTexts

2.2E: Exercises for Section 2.2

  • Page ID
    80051
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Intuitive Definition of Limits

    For exercises 1 - 2, consider the function \(f(x)=\dfrac{x^2−1}{|x−1|}\).

    1) [T] Complete the following table for the function. Round your solutions to four decimal places.

    \(x\) \(f(x)\) \(x\) \(f(x)\)
    0.9 a. 1.1 e.
    0.99 b. 1.01 f.
    0.999 c. 1.001 g.
    0.9999 d. 1.0001 h.

    2) What do your results in the preceding exercise indicate about the two-sided limit \(\displaystyle \lim_{x→1}f(x)\)? Explain your response.

    Answer:

    \(\displaystyle \lim_{x \to 1}f(x)\) does not exist because \(\displaystyle \lim_{x \to 1^−}f(x)=−2≠\lim_{x \to 1^+}f(x)=2\).

     

    For exercises 3 - 5, consider the function \(f(x)=(1+x)^{1/x}\).

    3) [T] Make a table showing the values of \(f\) for \(x=−0.01,\;−0.001,\;−0.0001,\;−0.00001\) and for \(x=0.01,\;0.001,\;0.0001,\;0.00001\). Round your solutions to five decimal places.

    \(x\) \(f(x)\) \(x\) \(f(x)\)
    -0.01 a. 0.01 e.
    -0.001 b. 0.001 f.
    -0.0001 c. 0.0001 g.
    -0.00001 d. 0.00001 h.

    4) What does the table of values in the preceding exercise indicate about the function \(f(x)=(1+x)^{1/x}\)?

    Answer:
    \(\displaystyle \lim_{x \to 0}(1+x)^{1/x}\approx 2.7183\).

    5) To which mathematical constant do the values in the preceding exercise appear to be approaching?  This is the actual limit here.

     

    In exercises 6 - 8, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.

    6) [T] \(\displaystyle \lim_{x \to 0}\frac{\sin 2x}{x};\quad ±0.1,\; ±0.01, \; ±0.001, \;±.0001\)

    \(x\) \(\frac{\sin 2x}{x}\) \(x\) \(\frac{\sin 2x}{x}\)
    -0.1 a. 0.1 e.
    -0.01 b. 0.01 f.
    -0.001 c. 0.001 g.
    -0.0001 d. 0.0001 h.
    Answer:
    a. 1.98669331; b. 1.99986667; c. 1.99999867; d. 1.99999999; e. 1.98669331; f. 1.99986667; g. 1.99999867; h. 1.99999999;
    \(\displaystyle \lim_{x \to 0}\frac{\sin 2x}{x}=2\)

    7) [T] \(\displaystyle \lim_{x \to 0}\frac{\sin 3x}{x} ±0.1, \; ±0.01, \; ±0.001, \; ±0.0001\)

    \(x\) \(\frac{\sin 3x}{x}\) \(x\) \(\frac{\sin 3x}{x}\)
    -0.1 a. 0.1 e.
    -0.01 b. 0.01 f.
    -0.001 c. 0.001 g.
    -0.0001 d. 0.0001 h.

    8) Use the preceding two exercises to conjecture (guess) the value of the following limit: \(\displaystyle \lim_{x \to 0}\frac{\sin ax}{x}\) for \(a\), a positive real value.

    Answer:
    \(\displaystyle \lim_{x \to 0}\frac{\sin ax}{x}=a\)

     

    [T] In exercises 9 - 12, set up a table of values to find the indicated limit. Round to eight digits.

    9) \(\displaystyle \lim_{x \to 2}\frac{x^2−4}{x^2+x−6}\)

    \(x\) \(\frac{x^2−4}{x^2+x−6}\) \(x\) \(\frac{x^2−4}{x^2+x−6}\)
    1.9 a. 2.1 e.
    1.99 b. 2.01 f.
    1.999 c. 2.001 g.
    1.9999 d. 2.0001 h.

    10) \(\displaystyle \lim_{x \to 1}(1−2x)\)

    \(x\) \(1−2x\) \(x\) \(1−2x\)
    0.9 a. 1.1 e.
    0.99 b. 1.01 f.
    0.999 c. 1.001 g.
    0.9999 d. 1.0001 h.
    Answer:
    a. −0.80000000; b. −0.98000000; c. −0.99800000; d. −0.99980000; e. −1.2000000; f. −1.0200000; g. −1.0020000; h. −1.0002000;
    \( \displaystyle \lim_{x \to 1}(1−2x)=−1\)

    11) \(\displaystyle \lim_{x \to 0}\frac{5}{1−e^{1/x}}\)

    \(x\) \(\frac{5}{1−e^{1/x}}\) \(x\) \(\frac{5}{1−e^{1/x}}\)
    -0.1 a. 0.1 e.
    -0.01 b. 0.01 f.
    -0.001 c. 0.001 g.
    -0.0001 d. 0.0001 h.

    12) \(\displaystyle \lim_{x \to 2}\frac{1−\frac{2}{x}}{x^2−4}\)

    \(x\) \(\frac{1−\frac{2}{x}}{x^2−4}\) \(x\) \(\frac{1−\frac{2}{x}}{x^2−4}\)
    1.9 a. 2.1 e.
    1.99 b. 2.01 f.
    1.999 c. 2.001 g.
    1.9999 d. 2.0001 h.
    Answer:
    a. 0.13495277; b. 0.12594300; c. 0.12509381; d. 0.12500938; e. 0.11614402; f. 0.12406794; g. 0.12490631; h. 0.12499063;
    \( \displaystyle ∴\lim_{x \to 2}\frac{1−\frac{2}{x}}{x^2−4}=0.1250=\frac{1}{8}\)


    In exercises 13 - 17, 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).

    13) \(\displaystyle \lim_{x→1^−}f(x)\)

    14) \(\displaystyle \lim_{x→1^+}f(x)\)

    Answer:
    \(2\)

    15) \(\displaystyle \lim_{x→1}f(x)\)

    16) \(\displaystyle \lim_{x→2}f(x)\)

    Answer:
    \(1\)

    17) \(f(1)\)

     

    In exercises 18 - 21, 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).

    18) \(\displaystyle \lim_{x→0^−}f(x)\)

    Answer:
    \(1\)

    19) \(\displaystyle \lim_{x→0^+}f(x)\)

    20) \(\displaystyle \lim_{x→0}f(x)\)

    Answer:
    DNE

    21) \(\displaystyle \lim_{x→2}f(x)\)

     

    In exercises 22 - 27, 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).

    22) \(\displaystyle \lim_{x→−2^−}f(x)\)

    Answer:
    \(0\)

    23) \(\displaystyle \lim_{x→−2^+}f(x)\)

    24) \(\displaystyle \lim_{x→−2}f(x)\)

    Answer:
    DNE

    25) \(\displaystyle \lim_{x→2^−}f(x)\)

    26) \(\displaystyle \lim_{x→2^+}f(x)\)

    Answer:
    \(2\)

    27) \(\displaystyle \lim_{x→2}f(x)\)

     

    In exercises 28 - 30, 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).

    28) \(\displaystyle \lim_{x→0^−}g(x)\)

    Answer:
    \(3\)

    29) \(\displaystyle \lim_{x→0^+}g(x)\)

    30) \(\displaystyle \lim_{x→0}g(x)\)

    Answer:
    DNE

     

    In exercises 31 - 33, 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.

    31) \(\displaystyle \lim_{x→0^−}h(x)\)

    32) \(\displaystyle \lim_{x→0^+}h(x)\)

    Answer:
    \(0\)

    33) \(\displaystyle \lim_{x→0}h(x)\)

     

    In exercises 34 - 38, 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.

    34) \(\displaystyle \lim_{x→0^−}f(x)\)

    Answer:
    \(-2\)

    35) \(\displaystyle \lim_{x→0^+}f(x)\)

    36) \(\displaystyle \lim_{x→0}f(x)\)

    Answer:
    DNE

    37) \(\displaystyle \lim_{x→1}f(x)\)

    38) \(\displaystyle \lim_{x→2}f(x)\)

    Answer:
    \(0\)

     

    39) 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

    40) 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.

     

    Contributors and Attributions

    • 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.


    This page titled 2.2E: Exercises for Section 2.2 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.