AP Calc AB Limits and Continuity

Question 1

What is lim(x→2) (x² − 4)/(x − 2)?

Accepted Answer

4

Answer

Factor the numerator as (x − 2)(x + 2) and cancel the common factor. That leaves lim(x→2) (x + 2) = 4.

Question 2

What is lim(x→∞) (3x² + 5x)/(2x² − 1)?

Accepted Answer

3/2

Answer

The numerator and denominator have the same degree, so the limit is the ratio of the leading coefficients, 3/2.

Question 3

Let f(x) = kx + 1 for x ≤ 3 and f(x) = x² − 5 for x > 3. For what value of k is f continuous at x = 3?

Accepted Answer

1

Answer

The right-hand limit at 3 is 3² − 5 = 4. The left piece must give 3k + 1 = 4, so k = 1.

Question 4

What is lim(x→0) sin(5x)/(2x)?

Accepted Answer

5/2

Answer

Rewrite the expression as (5/2)·[sin(5x)/(5x)]. Since sin(u)/u → 1 as u → 0, the limit is 5/2.

Question 5

The function f is continuous on the closed interval [1, 4], with f(1) = −3 and f(4) = 6. Which statement must be true?

Accepted Answer

f(c) = 0 for some c in (1, 4)

Answer

By the Intermediate Value Theorem, a continuous f takes every value between −3 and 6 on (1, 4). Since 0 lies between them, f(c) = 0 for some c in (1, 4).

AP Calc AB Practice Test

AP Calc AB Practice Test

AP Calc AB Limits and Continuity