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AP Calculus BC Practice Quiz: Selecting Procedures for Determining Limits

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

Which of the following is the most direct method to evaluate the limit of f(x) = (x^2 + 2x - 3) / (x + 1) as x approaches 2?

All Questions (7)

Which of the following is the most direct method to evaluate the limit of f(x) = (x^2 + 2x - 3) / (x + 1) as x approaches 2?

A) Factoring the numerator and denominator

B) Applying L'Hôpital's Rule

C) Direct substitution

D) Multiplying by the conjugate

Correct Answer: C

The function f(x) is a rational function. The first step in evaluating the limit of a rational function is to try direct substitution. Since the denominator is not zero when x = 2 (2 + 1 = 3), the function is continuous at x = 2, and the limit can be found by directly substituting the value of x into the function. Other methods like factoring or L'Hôpital's Rule are unnecessary.

To evaluate the limit of f(x) = (x^2 - 9) / (x - 3) as x approaches 3, direct substitution results in the indeterminate form 0/0. What is the most appropriate next step to determine the limit?

A) Conclude that the limit does not exist.

B) Apply the Squeeze Theorem.

C) Factor the numerator and simplify the expression.

D) Analyze the limit as x approaches infinity.

Correct Answer: C

When direct substitution yields the indeterminate form 0/0, it suggests that the numerator and denominator share a common factor. In this case, the numerator x^2 - 9 can be factored as (x - 3)(x + 3). The (x - 3) term can then be canceled from the numerator and denominator, removing the discontinuity and allowing the limit to be evaluated by substitution into the simplified expression.

Which procedure is necessary to find the limit of f(x) = (sqrt(x + 4) - 2) / x as x approaches 0?

A) Direct substitution, as the function is continuous at x=0.

B) Dividing the numerator and denominator by the highest power of x.

C) Multiplying the numerator and denominator by the conjugate of the numerator.

D) Using the special trigonometric limit for sin(x)/x.

Correct Answer: C

Direct substitution results in the indeterminate form 0/0. The presence of a square root in the expression suggests that multiplying by the conjugate is the appropriate technique. Multiplying the numerator and denominator by (sqrt(x + 4) + 2) will rationalize the numerator, allowing for simplification that resolves the indeterminate form.

The evaluation of the limit of f(x) = (3x^2 - 5x + 2) / (7x^2 + x - 1) as x approaches infinity requires which of the following procedures?

A) Factoring both the numerator and the denominator.

B) Comparing the degrees of the numerator and denominator.

C) Applying L'Hôpital's Rule repeatedly until the limit is found.

D) Substituting a very large number for x to approximate the limit.

Correct Answer: B

To find the limit of a rational function as x approaches infinity, the most efficient method is to compare the degrees of the polynomial in the numerator and the denominator. Since the degrees are the same (both are 2), the limit is the ratio of the leading coefficients, which is 3/7. An alternative, but equivalent, procedure is to divide every term by the highest power of x in the denominator (x^2), which also leads to the same result.

Consider the limit of f(x) = (1 - cos(x)) / x as x approaches 0. Direct substitution yields 0/0. Which of the following justifies the use of L'Hôpital's Rule as a valid procedure to find this limit?

A) The function f(x) is a rational function.

B) The numerator and denominator are both polynomials.

C) The limit results in an indeterminate form (0/0), and the numerator and denominator are both differentiable functions.

D) The function involves a trigonometric expression.

Correct Answer: C

L'Hôpital's Rule can be applied to a limit of the form f(x)/g(x) if it results in an indeterminate form like 0/0 or ∞/∞, and both f(x) and g(x) are differentiable near the point the limit is approaching. Here, the limit is 0/0, and both 1 - cos(x) and x are differentiable, so the conditions for L'Hôpital's Rule are met.

To determine the limit of the piecewise function f(x) defined as f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, as x approaches 1, what must be done?

A) Use the definition for x < 1 because the limit approaches from the left.

B) Use the definition for x ≥ 1 because the function is defined at x=1.

C) Evaluate the one-sided limits as x approaches 1 from the left and from the right and compare them.

D) Apply L'Hôpital's Rule since there is a break in the function.

Correct Answer: C

For a limit to exist at a point where a piecewise function changes its definition, the limit from the left must equal the limit from the right. Therefore, one must calculate the limit as x approaches 1- (using x^2) and the limit as x approaches 1+ (using 2x - 1). If these two one-sided limits are equal, the overall limit exists and is equal to that value.

Which of the following limit problems is best solved using the Squeeze Theorem?

A) The limit of (x^3 - 1) / (x - 1) as x approaches 1

B) The limit of x^2 * sin(1/x) as x approaches 0

C) The limit of (5x^4 - 2x) / (3x^4 + 1) as x approaches infinity

D) The limit of |x - 2| / (x - 2) as x approaches 2

Correct Answer: B

The Squeeze Theorem is ideal for functions that are trapped or 'squeezed' between two other functions that have the same limit. The function x^2 * sin(1/x) contains the oscillating term sin(1/x), which is always between -1 and 1. By multiplying the inequality -1 ≤ sin(1/x) ≤ 1 by x^2, we get -x^2 ≤ x^2 * sin(1/x) ≤ x^2. Since the limits of both -x^2 and x^2 as x approaches 0 are 0, the Squeeze Theorem states that the limit of x^2 * sin(1/x) must also be 0. The other options are best solved by factoring (A), comparing degrees (C), and evaluating one-sided limits (D).