Determining Limits Using Algebraic Manipulation Practice

Sample questions

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1. Sample 1difficulty 2/5

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

   • A

     $4$

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   • B

     $2$

   • C

     Does not exist

   • D

     $0$

   Why

   Factor the numerator: $(x-2)(x+2)/(x-2) = x + 2$ for $x \ne 2$. The limit is $2 + 2 = 4$.

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2. Sample 2difficulty 2/5

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

   • A

     $1$

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   • B

     $-1$

   • C

     $2$

   • D

     $0$

   Why

   $x^2 + 3x + 2 = (x+1)(x+2)$. Cancel: $\lim_{x \to -1}(x+2) = 1$.

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3. Sample 3difficulty 2/5

   Evaluate $\displaystyle\lim_{x\to 5}\dfrac{x^2 - 25}{x^2 - 6x + 5}$.

   • A

     $1$

   • B

     Does not exist

   • C

     $\tfrac{5}{2}$

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   • D

     $\tfrac{2}{5}$

   Why

   Top: $(x-5)(x+5)$. Bottom: $(x-5)(x-1)$. Cancel: $(x+5)/(x-1)$. At $x = 5$: $10/4 = 5/2$.

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4. Sample 4difficulty 3/5

   Evaluate $\displaystyle\lim_{x\to 0}\dfrac{(x+2)^2 - 4}{x}$.

   • A

     $2$

   • B

     $\infty$

   • C

     $4$

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   • D

     $0$

   Why

   Expand: $(x+2)^2 - 4 = x^2 + 4x + 4 - 4 = x^2 + 4x$. Divide by $x$: $x + 4 \to 4$.

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5. Sample 5difficulty 3/5

   Evaluate $\displaystyle\lim_{x\to 2}\dfrac{x^3 - 8}{x - 2}$.

   • A

     $0$

   • B

     $8$

   • C

     $12$

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   • D

     $4$

   Why

   $x^3 - 8 = (x-2)(x^2 + 2x + 4)$. Cancel: $x^2 + 2x + 4 \to 4 + 4 + 4 = 12$.

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