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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→∞3x2+1x2−4\lim_{x\to\infty}\frac{3x^2+1}{x^2-4}.

Please select an answer first.

Correct answer: A

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: Equal degrees give the ratio of leading coefficients, 3/1=3.
  3. Check continuity or growth: The correct answer is 3.

Equal degrees give the ratio of leading coefficients, 3/1=3.

2. QuestionMultiple choice

Evaluate lim⁡x→∞5x+2x2+1\lim_{x\to\infty}\frac{5x+2}{x^2+1}.

Please select an answer first.

Correct answer: B

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: The denominator has higher degree, so the rational function approaches zero.
  3. Check continuity or growth: The correct answer is 0.

The denominator has higher degree, so the rational function approaches zero.

3. QuestionMultiple choice

What is lim⁡x→0x2sin⁡(1x)\lim_{x\to0}x^2\sin\left(\frac{1}{x}\right)?

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: Because -1≤sin(1/x)≤1, multiplying by x^2 squeezes the expression to 0.
  3. Check continuity or growth: The correct answer is 0.

Because -1≤sin(1/x)≤1, multiplying by x^2 squeezes the expression to 0.

4. QuestionMultiple choice

Which theorem supports the result in lim⁡x→0x2sin⁡(1x)\lim_{x\to0}x^2\sin(\frac{1}{x})?

Please select an answer first.

Correct answer: D

The correct answer is The Squeeze Theorem.

  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 bounded between -x^2 and x^2, both of which approach zero.
  3. Check continuity or growth: The correct answer is The Squeeze Theorem.

The expression is bounded between -x^2 and x^2, both of which approach zero.

5. QuestionMultiple choice

What is the horizontal asymptote of f(x)=2x2+1x2+7f(x)=\frac{2x^2+1}{x^2+7}?

Please select an answer first.

Correct answer: A

The correct answer is y=2y=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 limit at positive or negative infinity is the ratio of leading coefficients, 2.
  3. Check continuity or growth: The correct answer is y=2y=2.

The limit at positive or negative infinity is the ratio of leading coefficients, 2.

6. QuestionMultiple choice

What is lim⁡x→∞(4x−1)\lim_{x\to\infty}(4x-1)?

Please select an answer first.

Correct answer: B

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: A positive linear expression grows without bound as x increases.
  3. Check continuity or growth: The correct answer is +infinity.

A positive linear expression grows without bound as x increases.

7. QuestionMultiple choice

What is lim⁡x→−∞(4x−1)\lim_{x\to-\infty}(4x-1)?

Please select an answer first.

Correct answer: C

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: A positive multiple of x decreases without bound as x approaches negative infinity.
  3. Check continuity or growth: The correct answer is -infinity.

A positive multiple of x decreases without bound as x approaches negative infinity.

8. QuestionMultiple choice

Which limit is used to define the instantaneous rate of change?

Please select an answer first.

Correct answer: D

The correct answer is lim⁡h→0f(x+h)−f(x)h\lim_{h\to0}\frac{f(x+h)-f(x)}{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 difference quotient with h approaching zero defines the derivative.
  3. Check continuity or growth: The correct answer is lim⁡h→0f(x+h)−f(x)h\lim_{h\to0}\frac{f(x+h)-f(x)}{h}.

The difference quotient with h approaching zero defines the derivative.

9. QuestionMultiple choice

If lim⁡x→af(x)=L\lim_{x\to a}f(x)=L and gg is continuous at L, what is lim⁡x→ag(f(x))\lim_{x\to a}g(f(x))?

Please select an answer first.

Correct answer: A

The correct answer is g(L)g(L).

  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 laws for continuous functions allow the outer function to be evaluated at the inner limit.
  3. Check continuity or growth: The correct answer is g(L)g(L).

The limit laws for continuous functions allow the outer function to be evaluated at the inner limit.

10. QuestionMultiple choice

What does a finite two-sided limit at x=a tell us by itself?

Please select an answer first.

Correct answer: B

The correct answer is Nearby outputs approach one 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 limit describes local behavior; it does not alone guarantee the function value or differentiability.
  3. Check continuity or growth: The correct answer is Nearby outputs approach one finite value.

A limit describes local behavior; it does not alone guarantee the function value or differentiability.

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