Asymptotes and Limits at Infinity Practice

Sample questions

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1. Sample 1difficulty 2/5

   Evaluate $\displaystyle\lim_{x \to \infty}\dfrac{2x^3 - 1}{x^2 + 5x}$.

   • A

     $\infty$

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   • B

     $0$

   • C

     $2$

   • D

     $-\infty$

   Why

   Numerator degree $3$ > denominator degree $2$. As $x \to \infty$, leading term $2x^3$ dominates → $+\infty$.

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2. Sample 2difficulty 2/5

   Evaluate $\displaystyle\lim_{x \to 2^+}\dfrac{1}{x - 2}$.

   • A

     $-\infty$

   • B

     $1$

   • C

     $+\infty$

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   • D

     $0$

   Why

   As $x \to 2^+$, $x - 2 \to 0^+$, so $1/(x-2) \to +\infty$.

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3. Sample 3difficulty 2/5

   Evaluate $\displaystyle\lim_{x\to\infty}\dfrac{x^4 - x}{2x^4 + 3}$.

   • A

     $0$

   • B

     $1$

   • C

     $\infty$

   • D

     $\tfrac{1}{2}$

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   Why

   Same degree (4); leading coefficient ratio: $1/2$.

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4. Sample 4difficulty 2/5

   Evaluate $\displaystyle\lim_{x \to -\infty}\dfrac{5x^3 + 1}{x^3 - 2x}$.

   • A

     $5$

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   • B

     $-5$

   • C

     $-\infty$

   • D

     $0$

   Why

   Same degree: ratio of leading coefficients = $5/1 = 5$ regardless of $x \to \pm\infty$.

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5. Sample 5difficulty 2/5

   As $x \to \infty$, the curve approaches the dashed horizontal line $y = 2$. So

   • A

     $\lim_{x\to\infty} f(x) = \infty$

   • B

     Limit doesn't exist

   • C

     $\lim_{x\to\infty} f(x) = 2$

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   • D

     $\lim_{x\to\infty} f(x) = 0$

   Why

   Horizontal asymptote at $y = 2$ → limit $= 2$.

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