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Mathematics LibreTexts

2.4E: Infinite Limits EXERCISES

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    10236

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    2.4: Infinite Limit Exercises

    In the following exercises, find the limit.

    In the following exercises, 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.

    J46) \(\displaystyle \lim_{x→10}f(x)=0\)

    J47) \(\displaystyle \lim_{x→−2^+}f(x)=3\)

    Answer:

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

    J48) \(\displaystyle \lim_{x→−8}f(x)=f(−8)\)

    J49) \(\displaystyle \lim_{x→6}f(x)=5\)

    Answer:

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

    J2.4.1)

    a. \(\displaystyle \lim_{x→−3^+}\frac{x}{x+3}\)

    b. \(\displaystyle \lim_{x→−3^-}\frac{x}{x+3}\)

    c. \(\displaystyle \lim_{x→−3}\frac{x}{x+3}\)


    Answer:
    a. −∞
    b. ∞
    c. DNE

    J2.4.2) \(\displaystyle \lim_{x→0}\ln |x|\)

    J2.4.3)

    a. \(\displaystyle \lim_{x→5^+}\frac{2}{x-5}\)

    b. \(\displaystyle \lim_{x→5^-}\frac{2}{x-5}\)

    c. \(\displaystyle \lim_{x→5}\frac{2}{x-5}\)

    Answer:
    a. ∞
    b. −∞
    c. DNE

    J2.4.4)

    a. \(\displaystyle \lim_{x→-2^+}\frac{x}{(x+2)^2}\)

    b. \(\displaystyle \lim_{x→-2^-}\frac{x}{(x+2)^2}\)

    c. \(\displaystyle \lim_{x→-2}\frac{x}{(x+2)^2}\)

    J2.4.5)

    a. \(\displaystyle \lim_{x→6^+}\frac{x}{(6-x)^2}\)

    b. \(\displaystyle \lim_{x→6^-}\frac{x}{(6-x)^2}\)

    c. \(\displaystyle \lim_{x→6}\frac{x}{(6-x)^2}\)

    Answer:
    a. ∞
    b. ∞
    c. ∞

    J2.4.6)

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

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

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

    J2.4.7) \(\displaystyle \lim_{x→1}\frac{x^3−1}{x^2−1}\)

    Answer:
    \(\displaystyle lim_{x→1}\frac{x^3−1}{x^2−1}=\displaystyle \lim_{x→1}\frac{(x-1)(x^2+x+1)}{(x-1)(x+1)}=\displaystyle \lim_{x→1}\frac{x^2+x+1}{x+1}=\frac{3}{2}\)

    J2.4.8) \(\displaystyle \lim_{x→1/2}\frac{2x}{2x−1}\)

    J2.4.9) \(\displaystyle \lim_{x→1/2}\frac{2x^2+3x−2}{2x−1}\)

    Answer:
    \(\displaystyle \lim_{x→ 1/2}\frac{2x^2+3x−2}{2x−1}=\displaystyle \lim_{x→1/2}\frac{(2x−1)(x+2)}{2x−1}=\displaystyle \lim_{x→1/2}(x+2)=\frac{5}{2}\)

    State the vertical asymptote for each function, if any.

    J2.4.10) \(f(x)=\ln x\)

    J2.4.11) \(g(x)=\frac{x+5}{x-4}\)

    Answer:
    \(x=4\)

    J2.4.12) \(g(x)=\frac{7}{x+5}\)

    J2.4.13) \(g(x)=\frac{7}{x}\)

    Answer:
    \(x=0\)

    J2.4.14)

    a. \(\displaystyle \lim_{x→\frac{\pi}{2}^+}\tan x=\)

    b. \(\displaystyle \lim_{x→\frac{\pi}{2}^-}\tan x=\)

    c. \(\displaystyle \lim_{x→\frac{\pi}{2}}\tan x=\)

    d. Does \(f(x)=\tan x\) have a vertical asymptote at \(x=\frac{\pi}{2}\)?