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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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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→2x2−4x−2\lim_{x\to2}\frac{x^2-4}{x-2}.

Please select an answer first.

Correct answer: A

The correct answer is 4.

  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: Factor x^2-4=(x-2)(x+2), cancel, and evaluate x+2 at 2.
  3. Check continuity or growth: The correct answer is 4.

Factor x^2-4=(x-2)(x+2), cancel, and evaluate x+2 at 2.

2. QuestionMultiple choice

Evaluate lim⁡x→3x2−9x−3\lim_{x\to3}\frac{x^2-9}{x-3}.

Please select an answer first.

Correct answer: B

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: Factor and cancel: (x^2-9)/(x-3)=x+3, whose limit at 3 is 6.
  3. Check continuity or growth: The correct answer is 6.

Factor and cancel: (x^2-9)/(x-3)=x+3, whose limit at 3 is 6.

3. QuestionMultiple choice

Evaluate lim⁡x→1x3−1x−1\lim_{x\to1}\frac{x^3-1}{x-1}.

Please select an answer first.

Correct answer: C

The correct answer is 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: Use x^3-1=(x-1)(x^2+x+1), then substitute x=1.
  3. Check continuity or growth: The correct answer is 3.

Use x^3-1=(x-1)(x^2+x+1), then substitute x=1.

4. QuestionMultiple choice

Evaluate lim⁡x→4x−2x−4\lim_{x\to4}\frac{\sqrt{x}-2}{x-4}.

Please select an answer first.

Correct answer: D

The correct answer is 14\frac{1}{4}.

  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: Rationalizing gives 1/(√x+2), which approaches 1/4.
  3. Check continuity or growth: The correct answer is 14\frac{1}{4}.

Rationalizing gives 1/(√x+2), which approaches 1/4.

5. QuestionMultiple choice

Evaluate lim⁡x→0sin⁡xx\lim_{x\to0}\frac{\sin x}{x}.

Please select an answer first.

Correct answer: A

The correct answer is 1.

  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: This is a fundamental trigonometric limit measured in radians.
  3. Check continuity or growth: The correct answer is 1.

This is a fundamental trigonometric limit measured in radians.

6. QuestionMultiple choice

What is the first algebraic step for lim⁡x→af(x)−f(a)x−a\lim_{x\to a}\frac{f(x)-f(a)}{x-a}?

Please select an answer first.

Correct answer: B

The correct answer is Recognize it as the difference quotient.

  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 expression is the limit definition of the derivative at x=a.
  3. Check continuity or growth: The correct answer is Recognize it as the difference quotient.

The expression is the limit definition of the derivative at x=a.

7. QuestionMultiple choice

Evaluate lim⁡x→01−cos⁡xx2\lim_{x\to0}\frac{1-\cos x}{x^2}.

Please select an answer first.

Correct answer: C

The correct answer is 12\frac{1}{2}.

  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: Using 1-cos x=2sin^2(x/2) and sin u/u→1 gives 1/2.
  3. Check continuity or growth: The correct answer is 12\frac{1}{2}.

Using 1-cos x=2sin^2(x/2) and sin u/u→1 gives 1/2.

8. QuestionMultiple choice

Why can a factored expression be useful in a 0/0 limit?

Please select an answer first.

Correct answer: D

The correct answer is It can reveal a common factor that represents a removable discontinuity.

  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: Canceling a common factor simplifies the nearby behavior while preserving the original excluded input.
  3. Check continuity or growth: The correct answer is It can reveal a common factor that represents a removable discontinuity.

Canceling a common factor simplifies the nearby behavior while preserving the original excluded input.

9. QuestionMultiple choice

Evaluate lim⁡x→−1x2−1x+1\lim_{x\to-1}\frac{x^2-1}{x+1}.

Please select an answer first.

Correct answer: A

The correct answer is -2.

  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: Factor x^2-1=(x-1)(x+1), cancel, and evaluate x-1 at -1.
  3. Check continuity or growth: The correct answer is -2.

Factor x^2-1=(x-1)(x+1), cancel, and evaluate x-1 at -1.

10. QuestionMultiple choice

If direct substitution gives 00\frac{0}{0}, what does it indicate?

Please select an answer first.

Correct answer: B

The correct answer is Further analysis is needed; the form is indeterminate.

  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 0/0 form does not determine the limit, so factoring, rationalizing, or another method may be needed.
  3. Check continuity or growth: The correct answer is Further analysis is needed; the form is indeterminate.

The 0/0 form does not determine the limit, so factoring, rationalizing, or another method may be needed.

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