PrepGo

AP Calculus AB 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 - 1) / (x + 5) 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 - 1) / (x + 5) as x approaches 2?

A) L'Hôpital's Rule

B) Factoring and canceling

C) Direct substitution

D) Multiplying by the conjugate

Correct Answer: C

The first step in evaluating a limit is always to try direct substitution. Since the function f(x) is a rational function and the denominator is not zero when x=2 (2+5=7), the function is continuous at x=2. Therefore, the limit can be found by directly substituting 2 for x.

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

A) Apply the Squeeze Theorem

B) Multiply the numerator and denominator by the conjugate

C) Conclude the limit does not exist

D) Factor the numerator and cancel the common factor

Correct Answer: D

When direct substitution results in 0/0 for a rational function, the next step is to try algebraic manipulation. The numerator, x^2 - 9, is a difference of squares and can be factored into (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.

Consider the limit of (√(x+4) - 2) / x as x approaches 0. After determining that direct substitution leads to an indeterminate form, which procedure should be applied to find the limit?

A) Multiplying the numerator and denominator by the conjugate of the numerator, √(x+4) + 2

B) Factoring the denominator

C) Applying L'Hôpital's Rule without checking conditions

D) Analyzing the limit from the left and right sides separately

Correct Answer: A

Direct substitution yields (√(4) - 2) / 0, which is 0/0. When an expression involves a radical and results in an indeterminate form, a common and effective technique is to multiply both the numerator and the denominator by the conjugate of the expression containing the radical. This often simplifies the expression and removes the indeterminate form.

What is the value of the limit of (e^x - 1) / sin(x) as x approaches 0?

A) 0

B) 1

C) e

D) The limit does not exist.

Correct Answer: B

Direct substitution of x=0 gives (e^0 - 1) / sin(0) = (1 - 1) / 0 = 0/0, which is an indeterminate form. Since the conditions for L'Hôpital's Rule are met (indeterminate form, differentiable functions), we can take the derivative of the numerator and the denominator. The derivative of e^x - 1 is e^x, and the derivative of sin(x) is cos(x). The new limit is lim (x→0) of e^x / cos(x), which evaluates to e^0 / cos(0) = 1/1 = 1.

Let f be the function defined by f(x) = { x^2 + 1, for x < 2; 2x + 1, for x > 2 }. To determine the limit of f(x) as x approaches 2, which procedure is necessary?

A) Use direct substitution with the first piece, x^2 + 1.

B) Apply L'Hôpital's Rule.

C) Evaluate the one-sided limits as x approaches 2 from the left and from the right and compare them.

D) Factor the expressions and look for common terms to cancel.

Correct Answer: C

Because the function is defined piecewise with the break at x=2, the overall limit as x approaches 2 exists only if the limit from the left equals the limit from the right. Therefore, one must evaluate lim (x→2⁻) f(x) using the rule for x < 2, and lim (x→2⁺) f(x) using the rule for x > 2. If these two values are equal, the limit exists.

When evaluating the limit of a rational function f(x) = (3x^4 - 2x) / (5x^4 + x^2) as x approaches infinity, what is the most efficient procedure?

A) Directly substitute infinity, which is undefined.

B) Divide every term in the numerator and denominator by the highest power of x in the denominator.

C) Factor the numerator and denominator.

D) Use L'Hôpital's Rule repeatedly until a determinate form is reached.

Correct Answer: B

For limits at infinity of rational functions, the most efficient method is to analyze the growth rates of the numerator and denominator. This can be done by dividing every term by the highest power of x in the expression (in this case, x^4). Alternatively, one can simply compare the degrees of the numerator and denominator. Since the degrees are equal (both are 4), the limit is the ratio of the leading coefficients, which is 3/5. Method B is the formal procedure that justifies this shortcut.

A student is asked to evaluate the limit of x^2 * cos(1/x) as x approaches 0. Direct substitution is inconclusive. Which theorem or rule is the most appropriate to correctly determine this limit?

A) L'Hôpital's Rule

B) The Squeeze Theorem

C) The Intermediate Value Theorem

D) The Mean Value Theorem

Correct Answer: B

This limit is a classic application of the Squeeze Theorem. We know that the cosine function is bounded between -1 and 1, so -1 ≤ cos(1/x) ≤ 1 for all x ≠ 0. Multiplying the inequality by x^2 (which is non-negative) gives -x^2 ≤ x^2 * cos(1/x) ≤ x^2. As x approaches 0, both -x^2 and x^2 approach 0. By the Squeeze Theorem, the function 'squeezed' between them, x^2 * cos(1/x), must also approach 0.