AP Calc BC Limits and Continuity

Question 1

What is lim(x→3) (x² − 9)/(x − 3)?

Accepted Answer

6

Answer

Factor the numerator as (x − 3)(x + 3) and cancel (x − 3). The limit is then lim(x→3) (x + 3) = 6.

Question 2

What is lim(x→0) sin(5x)/(2x)?

Accepted Answer

5/2

Answer

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

Question 3

Let f(x) = kx + 2 for x ≤ 2 and f(x) = x² − k for x > 2. For what value of k is f continuous at x = 2?

Accepted Answer

2/3

Answer

Continuity requires the left and right limits to equal f(2). So 2k + 2 = 4 − k, which gives k = 2/3.

Question 4

What is lim(x→∞) (3x² − 5x + 1)/(2x² + 7)?

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 5

What is lim(x→2⁻) 1/(x − 2)?

Accepted Answer

−∞

Answer

For x slightly less than 2, x − 2 is a small negative number, so 1/(x − 2) is a large negative number. The limit is −∞.

AP Calc BC Practice Test