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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

Which expression defines f'(a)?

Please select an answer first.

Correct answer: A

The correct answer is lim⁡h→0f(a+h)−f(a)h\lim_{h\to0}\frac{f(a+h)-f(a)}{h}.

  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 derivative at a is the limit of the difference quotient as the input change approaches zero.
  3. Check continuity or growth: The correct answer is lim⁡h→0f(a+h)−f(a)h\lim_{h\to0}\frac{f(a+h)-f(a)}{h}.

The derivative at a is the limit of the difference quotient as the input change approaches zero.

2. QuestionMultiple choice

What is the derivative of f(x)=x2f(x)=x^2?

Please select an answer first.

Correct answer: B

The correct answer is 2x2x.

  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: Apply the power rule: d(x^2)/dx=2x.
  3. Check continuity or growth: The correct answer is 2x2x.

Apply the power rule: d(x^2)/dx=2x.

3. QuestionMultiple choice

What is the derivative of a constant function f(x)=9f(x)=9?

Please select an answer first.

Correct answer: C

The correct answer is 0.

  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 constant has no change in output, so its rate of change is zero.
  3. Check continuity or growth: The correct answer is 0.

A constant has no change in output, so its rate of change is zero.

4. QuestionMultiple choice

What does f'(a) represent geometrically?

Please select an answer first.

Correct answer: D

The correct answer is The slope of the tangent line at x=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: The derivative is the instantaneous slope of the graph.
  3. Check continuity or growth: The correct answer is The slope of the tangent line at x=a.

The derivative is the instantaneous slope of the graph.

5. QuestionMultiple choice

What is the difference between average and instantaneous rate of change?

Please select an answer first.

Correct answer: A

The correct answer is Average uses an interval; instantaneous uses a limiting point.

  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 instantaneous rate is obtained by shrinking the interval to zero.
  3. Check continuity or growth: The correct answer is Average uses an interval; instantaneous uses a limiting point.

The instantaneous rate is obtained by shrinking the interval to zero.

6. QuestionMultiple choice

If position is s(t)=t2+3ts(t)=t^2+3t, what is velocity at t=2?

Please select an answer first.

Correct answer: B

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: Velocity is s'(t)=2t+3, so s'(2)=7.
  3. Check continuity or growth: The correct answer is 7.

Velocity is s'(t)=2t+3, so s'(2)=7.

7. QuestionMultiple choice

What is the derivative of f(x)=4x3−2xf(x)=4x^3-2x?

Please select an answer first.

Correct answer: C

The correct answer is 12x2−212x^2-2.

  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: Differentiate term by term using the power rule.
  3. Check continuity or growth: The correct answer is 12x2−212x^2-2.

Differentiate term by term using the power rule.

8. QuestionMultiple choice

If f'(x)>0 on an interval, what is f doing there?

Please select an answer first.

Correct answer: D

The correct answer is Increasing.

  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 positive derivative indicates increasing output as x increases.
  3. Check continuity or growth: The correct answer is Increasing.

A positive derivative indicates increasing output as x increases.

9. QuestionMultiple choice

If a differentiable function has a local minimum at x=a, what is usually true?

Please select an answer first.

Correct answer: A

The correct answer is f′(a)=0f'(a)=0.

  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 differentiable interior extremum has a horizontal tangent.
  3. Check continuity or growth: The correct answer is f′(a)=0f'(a)=0.

A differentiable interior extremum has a horizontal tangent.

10. QuestionMultiple choice

Which statement is true about differentiability?

Please select an answer first.

Correct answer: B

The correct answer is Differentiability at a point implies continuity there.

  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 differentiable function cannot have a jump, hole, or corner at that point.
  3. Check continuity or growth: The correct answer is Differentiability at a point implies continuity there.

A differentiable function cannot have a jump, hole, or corner at that point.

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