AP Calculus BC Unit 1 — Limits and Continuity Practice Test

Question 1

Which of the following functions have the same instantaneous rate of changewhen x=1 and x=2 ?

A. f ( x )=3 xB. f ( x )=x2

C. f ( x )=sin xD. f ( x )=x3

Question 2

At which point does the function have an approximate instantaneous rate ofchange of 0?

A. Point AB. Point BC. Point CD. Point D

Question 3

Which of the following functions has an instantaneous rate of change ofapproximately 2 when x=1 ?

A. f ( x )=cos ( x )

B. f ( x )=2 x2

C. f ( x )=3 xD. f ( x )=x2

Question 4

Where is the following function discontinuous?

A. I, II, and IIIB. I and II onlyC. I and III onlyD. II and III only

Question 5

For what x value does the instantaneous rate of change of f ( x ) not existabove?

A. x=−4B. x=−3C. x=1D. x=2

Question 6

Given the graph of f ( x ) above, what is limx→ 0

f ( x ) ?

A. 1B. 0C. ∞D. ‒∞E. Does not exist

Question 7

For the piecewise function:

,

find +¿

x→0¿ lim❑

❑ .

A.12

B. Does not exist.C. 1D. 0E. Any real number.

Question 8

Given the above graph of f ( x ) , what is +¿ f ( x )?

x→0¿ lim❑

❑

A. ‒∞

B. 0C. ‒1D. ∞E. 1

Question 9

The function f is defined over the real numbers. This table gives a few valuesof f .

What is a reasonable estimate for limx→ 3

f ( x ) ?

A. 2.9B. 3C. 9.9D. 10E. The limit doesn’t exist

Question 10

Functions h and g are graphed.

Find limx→ 0

(h (x ) −g ( x ) ) .

A. 0B. 2C. 3D. 5E. The limit doesn’t exist

Question 11

Functions h and g are graphed.

Find limx→ 0

h ( x )

g ( x ).

A. ‒4B. 0C. 1D. 4

E. The limit doesn’t exist

Question 12

cos ( x )=¿ limx→

π4

❑?

A.12

B. 1

C. √22

D. The limit doesn’t exist

Question 13

Find limx→− 2

x3+3 x2

+2 xx+2

.

A. 6B. 0C. 2D. The limit doesn’t exist

Question 14

Find limx→3

x −3

√4 x+4−4.

A. −4B. 1C. 2D. The limit doesn’t exist

Question 15

We want to find limx→ 0

xsin ( x )

. Direct substitution and other algebraic methods

don’t seem to work.

Looking at the graph of f ( x )=x

sin ( x ), we can estimate that the limit is equal to 1.

To prove that limx→0

xsin ( x )

=1 , we can use the squeeze theorem.

Madison suggested that we use the functions g ( x )=−|x|+0.5 and h ( x )=|x|+1.5in order to apply the squeeze theorem.

Does Madison’s suggestion seem to be correct?

A. Yes, Madison’s suggestion seems to be correct.B. No, Madison’s suggestion is incorrect because it’s not true that one function is

always below f and one function is always above it for x -values near 0.C. No, Madison’s suggestion is incorrect because it’s not true that the limits of g

and h are both equal to 1.

Question 16

Which of the following functions are continuous for all real numbers?

g ( x )=2x

f ( x )=ln ( x )

A. g onlyB. f onlyC. Both g and fD. Neither g nor f

Question 17

Consider graphs A, B, and C. The dashed lines represent asymptotes.

Which graphs agree with this statement?

limx →∞

h ( x )=0

Choose all answers that apply.

A.

B. correct answer

C.

Question 18

Let h be a continuous function on the closed interval [0,4], where h (0 )=2 andh (4 )=−2 .

Which of the following is guaranteed by the Intermediate Value Theorem?

A. h (c )=−1 for at least one c between −2 and 2B. h (c )=3 for at least one c between 0 and 4C. h (c )=3 for at least one c between −2 and 2D. h (c )=−1 for at least one c between 0 and 4

Question 19

Anastasia’s teacher gave her a flow chart (below) and asked her to find

limx →− 5

f ( x ) for f ( x )=x+5

√19−6 x−2.

Which shows Anastasia’s easiest path to finding the limit?

A. A→B→CB. B→CC. A→CD. A→D→HE. B→E→H

Answer Key

1. A2. B3. D4. D5. B6. E7. C8. D9. D10.E11.E12.C13.C14.C15.C16.A17.B18.D19.C