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Mathematics LibreTexts

2.5E: Limits at Infinity EXERCISES

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  • Page ID
    10190
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    2.5E: Limits at Infinity EXERCISES 

    For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.

    251)

    Answer:
    \(x=1\)

    252)

    253)

    Answer:
    \(x=−1,x=2\)

    254)

     

    255)

    Answer:
    \(x=0\)

     

    For the following functions \(f(x)\), determine whether there is an asymptote at \(x=a\). Justify your answer without graphing on a calculator.

    256) \(f(x)=\frac{x+1}{x^2+5x+4},a=−1\)

    257) \(f(x)=\frac{x}{x−2},a=2\)

    Answer:
    Yes, there is a vertical asymptote

    258) \(f(x)=(x+2)^{3/2},a=−2\)

    259) \(f(x)=(x−1)^{−1/3},a=1\)

    Answer:
    Yes, there is vertical asymptote

    260) \(f(x)=1+x^{−2/5},a=1\)

     

    For the following exercises, evaluate the limit.

    261) \(lim_{x→∞}\frac{1}{3x+6}\)

    Answer:
    \(0\)

    262) \(lim_{x→∞}\frac{2x−5}{4x}\)

    263) \(lim_{x→∞}\frac{x^2−2x+5}{x+2}\)

    Answer:
    \(∞\)

    264) \(lim_{x→−∞}\frac{3x^3−2x}{x^2+2x+8}\)

    265) \(lim_{x→−∞}\frac{x^4−4x^3+1}{2−2x^2−7x^4}\)

    Answer:
    \(−\frac{1}{7}\)

    266) \(lim_{x→∞}\frac{3x}{\sqrt{x^2+1}}\)

    267) \(lim_{x→−∞}\frac{\sqrt{4x2−1}}{x+2}\)

    Answer:
    \(−2\)

    268) \(lim_{x→∞}\frac{4x}{\sqrt{x2−1}}\)

    269) \(lim_{x→−∞}\frac{4x}{\sqrt{x2−1}}\)

    Answer:
    \(−4\)

    270) \(lim_{x→∞}\frac{2\sqrt{x}}{x−\sqrt{x}+1}\)

     

    For the following exercises, find the horizontal and vertical asymptotes.

    271) \(f(x)=x−\frac{9}{x}\)

    Answer:
    Horizontal: none, vertical: \(x=0\)

    272) \(f(x)=\frac{1}{1−x^2}\)

    273) \(f(x)=\frac{x^3}{4−x^2}\)

    Answer:
    Horizontal: none, vertical: \(x=±2\)

    274) \(f(x)=\frac{x^2+3}{x^2+1}\)

    275) \(f(x)=sin(x)sin(2x)\)

    Answer:
    Horizontal: none, vertical: none

    276) \(f(x)=cosx+cos(3x)+cos(5x)\)

    277) \(f(x)=\frac{xsin(x)}{x^2−1}\)

    Answer:
    Horizontal: \(y=0,\) vertical: \(x=±1\)

    278) \(f(x)=\frac{x}{sin(x)}\)

    279) \(f(x)=\frac{1}{x^3+x^2}\)

    Answer:
    Horizontal: \(y=0,\) vertical: \(x=0\) and \(x=−1\)

    280) \(f(x)=\frac{1}{x−1}−2x\)

    281) \(f(x)=\frac{x^3+1}{x^3−1}\)

    Answer:
    Horizontal: \(y=1,\) vertical: \(x=1\)

    282) \(f(x)=\frac{sinx+cosx}{sinx−cosx}\)

    283) \(f(x)=x−sinx\)

    Answer:
    Horizontal: none, vertical: none

    284) \(f(x)=\frac{1}{x}−\sqrt{x}\)

     

    For the following exercises, construct a function \(f(x)\) that has the given asymptotes.

    285) \(x=1\) and \(y=2\)

    Answer:
    \(x=0\)

    Solution: Answers will vary, for example: \(y=\frac{2x}{x−1}\)

    286) \(x=1\) and \(y=0\)

    287) \(y=4, x=−1\)

    Answer:
    \(x=0\)

    Solution: Answers will vary, for example: \(y=\frac{4x}{x+1}\)

    288) \(x=0\)

     

    CHAPTER REVIEW EXERCISES

    CR 1) \(lim_{x→∞}\frac{3x\sqrt{x^2+1}}{\sqrt{x4−1}}\)

    Answer:
    \(3\)

    CR 2) \(lim_{x→∞}cos(\frac{1}{x})\)

    CR 3) \(lim_{x→1}\frac{x−1}{sin(πx)}\)

    Answer:
    \(−\frac{1}{π}\)

    CR 4) \(lim_{x→∞}(3x)^{1/x}\)