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Mathematics LibreTexts

2.5E: Exercises for Section 2.5

  • Page ID
    50407
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    In exercises 1 - 4, write the appropriate \(ε−δ\) definition for each of the given statements.

    1) \(\displaystyle \lim_{x→a}f(x)=N\)

    2) \(\displaystyle \lim_{t→b}g(t)=M\)

    Answer
    For every \(ε>0\), there exists a \(δ>0\), so that if \(0<|t−b|<δ\), then \(|g(t)−M|<ε\)

    3) \(\displaystyle \lim_{x→c}h(x)=L\)

    4) \(\displaystyle \lim_{x→a}φ(x)=A\)

    Answer
    For every \(ε>0\), there exists a \(δ>0\), so that if \(0<|x−a|<δ\), then \(|φ(x)−A|<ε\)

    The following graph of the function \(f\) satisfies \(\displaystyle \lim_{x→2}f(x)=2\). In the following exercises, determine a value of \(δ>0\) that satisfies each statement.

    A function drawn in quadrant one for x > 0. It is an increasing concave up function, with points approximately (0,0), (1, .5), (2,2), and (3,4).

    5) If \(0<|x−2|<δ\), then \(|f(x)−2|<1\).

    6) If \(0<|x−2|<δ\), then \(|f(x)−2|<0.5\).

    Answer
    \(δ≤0.25\)

    The following graph of the function \(f\) satisfies \(\displaystyle \lim_{x→3}f(x)=−1\). In the following exercises, determine a value of \(δ>0\) that satisfies each statement.

    A graph of a decreasing linear function, with points (0,2), (1,1), (2,0), (3,-1), (4,-2), and so on for x >= 0.

    7) If \(0<|x−3|<δ\), then \(|f(x)+1|<1\).

    8) If \(0<|x−3|<δ\), then \(|f(x)+1|<2\).

    Answer
    \(δ≤2\)

    The following graph of the function \(f\) satisfies \(\displaystyle \lim_{x→3}f(x)=2\). In the following exercises, for each value of \(ε\), find a value of \(δ>0\) such that the precise definition of limit holds true.

    A graph of an increasing linear function intersecting the x axis at about (2.25, 0) and going through the points (3,2) and, approximately, (1,-5) and (4,5).

    9) \(ε=1.5\)

    10) \(ε=3\)

    Answer
    \(δ≤1\)

    [T] In exercises 11 - 12, use a graphing calculator to find a number \(δ\) such that the statements hold true.

    11) \(\left|\sin(2x)−\frac{1}{2}\right|<0.1\), whenever \(\left|x−\frac{π}{12}\right|<δ\)

    12) \(\left|\sqrt{x−4}−2\right|<0.1\), whenever \(|x−8|<δ\)

    Answer
    \(δ<0.3900\)

    In exercises 13 - 17, use the precise definition of limit to prove the given limits.

    13) \(\displaystyle \lim_{x→2}\,(5x+8)=18\)

    14) \(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3}=6\)

    Answer
    Let \(δ=ε\). If \(0<|x−3|<ε\), then \(\left|\dfrac{x^2−9}{x−3} - 6\right| = \left|\dfrac{(x+3)(x−3)}{x−3} - 6\right| = |x+3−6|=|x−3|<ε\).

    15) \(\displaystyle \lim_{x→2}\frac{2x^2−3x−2}{x−2}=5\)

    16) \(\displaystyle \lim_{x→0}x^4=0\)

    Answer
    Let \(δ=\sqrt[4]{ε}\). If \(0<|x|<\sqrt[4]{ε}\), then \(\left|x^4-0\right|=x^4<ε\).

    17) \(\displaystyle \lim_{x→2}\,(x^2+2x)=8\)

    In exercises 18 - 20, use the precise definition of limit to prove the given one-sided limits.

    18) \(\displaystyle \lim_{x→5^−}\sqrt{5−x}=0\)

    Answer
    Let \(δ=ε^2\). If \(-ε^2 < x - 5 < 0,\) we can multiply through by \(-1\) to get \(0<5-x<ε^2.\)
    Then \(\left|\sqrt{5−x} - 0\right|=\sqrt{5−x} < \sqrt{ε^2} = ε\).

    19) \(\displaystyle \lim_{x→0^+}f(x)=−2\), where \(f(x)=\begin{cases}8x−3, & \text{if }x<0\\4x−2, & \text{if }x≥0\end{cases}\).

    20) \(\displaystyle \lim_{x→1^−}f(x)=3\), where \(f(x)=\begin{cases}5x−2, & \text{if }x<1\\7x−1, & \text{if }x≥1\end{cases}\).

    Answer
    Let \(δ=ε/5\). If \(−ε/5< x - 1<0,\) we can multiply through by \(-1\) to get \(0<1-x<ε/5.\)
    Then \(|f(x)−3|=|5x-2-3| = |5x−5| = 5(1-x),\) since \(x<1\) here.
    And \(5(1-x) < 5(ε/5) = ε\).

    In exercises 21 - 23, use the precise definition of limit to prove the given infinite limits.

    21) \(\displaystyle \lim_{x→0}\frac{1}{x^2}=∞\)

    22) \(\displaystyle \lim_{x→−1}\frac{3}{(x+1)^2}=∞\)

    Answer
    Let \(δ=\sqrt{\frac{3}{N}}\). If \(0<|x+1|<\sqrt{\frac{3}{N}}\), then \(f(x)=\frac{3}{(x+1)^2}>N\).

    23) \(\displaystyle \lim_{x→2}−\frac{1}{(x−2)^2}=−∞\)

    24) An engineer is using a machine to cut a flat square of Aerogel of area \(144 \,\text{cm}^2\). If there is a maximum error tolerance in the area of \(8 \,\text{cm}^2\), how accurately must the engineer cut on the side, assuming all sides have the same length? How do these numbers relate to \(δ\), \(ε\), \(a\), and \(L\)?

    Answer
    \(0.033 \text{ cm}, \,ε=8,\,δ=0.33,\,a=12,\,L=144\)

    25) Use the precise definition of limit to prove that the following limit does not exist: \(\displaystyle \lim_{x→1}\frac{|x−1|}{x−1}.\)

    26) Using precise definitions of limits, prove that \(\displaystyle \lim_{x→0}f(x)\) does not exist, given that \(f(x)\) is the ceiling function. (Hint: Try any \(δ<1\).)

    Answer
    Answers may very.

    27) Using precise definitions of limits, prove that \(\displaystyle \lim_{x→0}f(x)\) does not exist: \(f(x)=\begin{cases}1, & \text{if }x\text{ is rational}\\0, & \text{if }x\text{ is irrational}\end{cases}\). (Hint: Think about how you can always choose a rational number \(0<r<d\), but \(|f(r)−0|=1\).)

    28) Using precise definitions of limits, determine \(\displaystyle \lim_{x→0}f(x)\) for \(f(x)=\begin{cases}x, & \text{if }x\text{ is rational}\\0, & \text{if }x\text{ is irrational}\end{cases}\). (Hint: Break into two cases, \(x\) rational and \(x\) irrational.)

    Answer
    \(0\)

    29) Using the function from the previous exercise, use the precise definition of limits to show that \(\displaystyle \lim_{x→a}f(x)\) does not exist for \(a≠0\)

    For exercises 30 - 32, suppose that \(\displaystyle \lim_{x→a}f(x)=L\) and \(\displaystyle \lim_{x→a}g(x)=M\) both exist. Use the precise definition of limits to prove the following limit laws:

    30) \(\displaystyle \lim_{x→a}(f(x)−g(x))=L−M\)

    Answer
    \(f(x)−g(x)=f(x)+(−1)g(x)\)

    31) \(\displaystyle \lim_{x→a}[cf(x)]=cL\) for any real constant \(c\) (Hint: Consider two cases: \(c=0\) and \(c≠0\).)

    32) \(\displaystyle \lim_{x→a}[f(x)g(x)]=LM\). (Hint: \(|f(x)g(x)−LM|= |f(x)g(x) −f(x)M +f(x)M −LM| ≤|f(x)||g(x) −M| +|M||f(x)−L|.)\)

    Answer
    Answers may vary.

    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.

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