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Limits: Introduction Practice Tests

These Grade 12 Math practice tests introduce the language and tools of limits, including direct substitution, one-sided limits, algebraic simplification, continuity, asymptotes, the Squeeze Theorem, derivatives, and instantaneous rates of change.

Skills covered

  • Evaluating polynomial, rational, and constant limits by direct substitution and limit laws
  • Factoring, rationalizing, and recognizing indeterminate forms such as 0/0
  • Analyzing one-sided limits, holes, jumps, vertical asymptotes, and continuity
  • Using the Intermediate Value Theorem and Squeeze Theorem to reason about function behavior
  • Connecting the derivative definition with tangent slopes, velocity, and instantaneous rate of change

How to use these practice tests

First check whether direct substitution works. If it produces an indeterminate form, simplify algebraically or use a known theorem, then interpret the result using continuity, graph behavior, or rate of change.

For more Grade 12 Math practice, return to the Grade 12 Math category.

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1. Test

Grade 12 Limits: Introduction Practice Tests

Practice Grade 12 Math questions about direct and one-sided limits, algebraic limit methods, continuity, asymptotes, the Squeeze Theorem, derivatives, and rates of change.

Choose the best answer. Check whether substitution is valid, identify one-sided behavior, simplify indeterminate forms, and interpret the limit in context.

10 Questions
0/10 Questions0 %
Correct 0
Incorrect 0
Unanswered 10
1. QuestionMultiple choice

Evaluate lim⁡x→2(x2+3x−1)\lim_{x\to2}(x^2+3x-1).

Please select an answer first.

Correct answer: A

The correct answer is 9.

  1. Identify the limit behavior: Determine the approaching input, the relevant side, the function form, and whether direct substitution is valid.
  2. Apply the limit method: A polynomial is continuous, so substitute x=2: 4+6-1=9.
  3. Check continuity or growth: The correct answer is 9.

A polynomial is continuous, so substitute x=2: 4+6-1=9.

2. QuestionMultiple choice

Evaluate lim⁡x→3(2x+5)\lim_{x\to3}(2x+5).

Please select an answer first.

Correct answer: B

The correct answer is 11.

  1. Identify the limit behavior: Determine the approaching input, the relevant side, the function form, and whether direct substitution is valid.
  2. Apply the limit method: Substitute x=3 into the linear expression: 2(3)+5=11.
  3. Check continuity or growth: The correct answer is 11.

Substitute x=3 into the linear expression: 2(3)+5=11.

3. QuestionMultiple choice

Evaluate lim⁡x→1x+4x+2\lim_{x\to1}\frac{x+4}{x+2}.

Please select an answer first.

Correct answer: C

The correct answer is 53\frac{5}{3}.

  1. Identify the limit behavior: Determine the approaching input, the relevant side, the function form, and whether direct substitution is valid.
  2. Apply the limit method: The denominator is nonzero at x=1, so direct substitution gives 5/3.
  3. Check continuity or growth: The correct answer is 53\frac{5}{3}.

The denominator is nonzero at x=1, so direct substitution gives 5/3.

4. QuestionMultiple choice

What is lim⁡x→57\lim_{x\to5}7?

Please select an answer first.

Correct answer: D

The correct answer is 7.

  1. Identify the limit behavior: Determine the approaching input, the relevant side, the function form, and whether direct substitution is valid.
  2. Apply the limit method: The limit of a constant function is the constant itself.
  3. Check continuity or growth: The correct answer is 7.

The limit of a constant function is the constant itself.

5. QuestionMultiple choice

Evaluate lim⁡x→−2(x3+1)\lim_{x\to-2}(x^3+1).

Please select an answer first.

Correct answer: A

The correct answer is -7.

  1. Identify the limit behavior: Determine the approaching input, the relevant side, the function form, and whether direct substitution is valid.
  2. Apply the limit method: Substitute x=-2: (-2)^3+1=-8+1=-7.
  3. Check continuity or growth: The correct answer is -7.

Substitute x=-2: (-2)^3+1=-8+1=-7.

6. QuestionMultiple choice

If lim⁡x→4f(x)=6\lim_{x\to4}f(x)=6, what is lim⁡x→4[f(x)+3]\lim_{x\to4}[f(x)+3]?

Please select an answer first.

Correct answer: B

The correct answer is 9.

  1. Identify the limit behavior: Determine the approaching input, the relevant side, the function form, and whether direct substitution is valid.
  2. Apply the limit method: Limit laws allow the constant to be added: 6+3=9.
  3. Check continuity or growth: The correct answer is 9.

Limit laws allow the constant to be added: 6+3=9.

7. QuestionMultiple choice

If lim⁡x→2f(x)=3\lim_{x\to2}f(x)=3, what is lim⁡x→22f(x)\lim_{x\to2}2f(x)?

Please select an answer first.

Correct answer: C

The correct answer is 6.

  1. Identify the limit behavior: Determine the approaching input, the relevant side, the function form, and whether direct substitution is valid.
  2. Apply the limit method: Constant multiple law gives 2·3=6.
  3. Check continuity or growth: The correct answer is 6.

Constant multiple law gives 2·3=6.

8. QuestionMultiple choice

If lim⁡x→1f(x)=2\lim_{x\to1}f(x)=2 and lim⁡x→1g(x)=4\lim_{x\to1}g(x)=4, what is lim⁡x→1[f(x)+g(x)]\lim_{x\to1}[f(x)+g(x)]?

Please select an answer first.

Correct answer: D

The correct answer is 6.

  1. Identify the limit behavior: Determine the approaching input, the relevant side, the function form, and whether direct substitution is valid.
  2. Apply the limit method: The sum law gives 2+4=6.
  3. Check continuity or growth: The correct answer is 6.

The sum law gives 2+4=6.

9. QuestionMultiple choice

What does lim⁡x→af(x)=L\lim_{x\to a}f(x)=L mean?

Please select an answer first.

Correct answer: A

The correct answer is The outputs approach L as x approaches a.

  1. Identify the limit behavior: Determine the approaching input, the relevant side, the function form, and whether direct substitution is valid.
  2. Apply the limit method: A limit describes nearby output behavior and does not require the function to be defined at a.
  3. Check continuity or growth: The correct answer is The outputs approach L as x approaches a.

A limit describes nearby output behavior and does not require the function to be defined at a.

10. QuestionMultiple choice

Can a limit exist even if f(a) is not defined?

Please select an answer first.

Correct answer: B

The correct answer is Yes, if both sides approach the same finite value.

  1. Identify the limit behavior: Determine the approaching input, the relevant side, the function form, and whether direct substitution is valid.
  2. Apply the limit method: A removable hole can have a limit even though the function value at that input is missing.
  3. Check continuity or growth: The correct answer is Yes, if both sides approach the same finite value.

A removable hole can have a limit even though the function value at that input is missing.

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