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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

What is the definition of continuity at x=a?

Please select an answer first.

Correct answer: A

The correct answer is f(a)f(a) exists, lim⁡x→af(x)\lim_{x\to a}f(x) exists, and they are equal.

  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: All three conditions are required for continuity at the point.
  3. Check continuity or growth: The correct answer is f(a)f(a) exists, lim⁡x→af(x)\lim_{x\to a}f(x) exists, and they are equal.

All three conditions are required for continuity at the point.

2. QuestionMultiple choice

Which function is continuous for every real x?

Please select an answer first.

Correct answer: B

The correct answer is f(x)=x3−2x+1f(x)=x^3-2x+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: Polynomials are continuous on all real numbers.
  3. Check continuity or growth: The correct answer is f(x)=x3−2x+1f(x)=x^3-2x+1.

Polynomials are continuous on all real numbers.

3. QuestionMultiple choice

What type of discontinuity is a single missing point that could be filled?

Please select an answer first.

Correct answer: C

The correct answer is 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: A hole is removable when the two-sided limit is finite.
  3. Check continuity or growth: The correct answer is Removable discontinuity.

A hole is removable when the two-sided limit is finite.

4. QuestionMultiple choice

What type of discontinuity occurs when the one-sided limits are different finite values?

Please select an answer first.

Correct answer: D

The correct answer is Jump 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: Different left and right limits create a jump.
  3. Check continuity or growth: The correct answer is Jump discontinuity.

Different left and right limits create a jump.

5. QuestionMultiple choice

What happens at a vertical asymptote?

Please select an answer first.

Correct answer: A

The correct answer is Function values grow without bound near a vertical input 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: An infinite limit often indicates a vertical asymptote.
  3. Check continuity or growth: The correct answer is Function values grow without bound near a vertical input value.

An infinite limit often indicates a vertical asymptote.

6. QuestionMultiple choice

For what value of c is the piecewise function continuous at x=2 if f(x)=x+1 for x<2 and f(x)=cx for x≥2?

Please select an answer first.

Correct answer: B

The correct answer is c=32c=\frac{3}{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: The left limit is 3, so 2c=3 and c=3/2.
  3. Check continuity or growth: The correct answer is c=32c=\frac{3}{2}.

The left limit is 3, so 2c=3 and c=3/2.

7. QuestionMultiple choice

What does the Intermediate Value Theorem guarantee under its hypotheses?

Please select an answer first.

Correct answer: C

The correct answer is A continuous function takes every value between its endpoint values.

  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: Continuity on an interval guarantees that no intermediate output value is skipped.
  3. Check continuity or growth: The correct answer is A continuous function takes every value between its endpoint values.

Continuity on an interval guarantees that no intermediate output value is skipped.

8. QuestionMultiple choice

What is lim⁡x→2−1x−2\lim_{x\to2^-}\frac{1}{x-2}?

Please select an answer first.

Correct answer: D

The correct answer is -infinity.

  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: Approaching 2 from the left makes x-2 a small negative number, so the quotient decreases without bound.
  3. Check continuity or growth: The correct answer is -infinity.

Approaching 2 from the left makes x-2 a small negative number, so the quotient decreases without bound.

9. QuestionMultiple choice

What is lim⁡x→2+1x−2\lim_{x\to2^+}\frac{1}{x-2}?

Please select an answer first.

Correct answer: A

The correct answer is +infinity.

  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: Approaching 2 from the right makes x-2 a small positive number.
  3. Check continuity or growth: The correct answer is +infinity.

Approaching 2 from the right makes x-2 a small positive number.

10. QuestionMultiple choice

For a rational function, which input must be excluded from the domain?

Please select an answer first.

Correct answer: B

The correct answer is Any input that makes the denominator zero.

  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: Division by zero is undefined, even when a factor can later be canceled for limit analysis.
  3. Check continuity or growth: The correct answer is Any input that makes the denominator zero.

Division by zero is undefined, even when a factor can later be canceled for limit analysis.

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