Name: ____________________________________
Unit 1: Limits
Review Sheet
1. In your own words define what a limit is.
2. True or False. If 𝑓 is undefined at 𝑥 = 𝑐, then the limit of 𝑓(𝑥) as 𝑥 approaches 𝑐 does not exist.
3. Find the vertical and horizontal asymptotes for each of the following functions:
1
a. 𝑓(𝑥) = 𝑥 2 −4
b. 𝑓(𝑥) =
4𝑥+1
5𝑥
1−𝑥
c. 𝑔(𝑥) = 2𝑥 2 −5𝑥−3
d. ℎ(𝑥) =
e. 𝑓(𝑥) =
𝑥 2 −6𝑥−7
𝑥 2 −1
𝑥 2 −2𝑥
𝑥+1
4. Sketch the function and state the domain and range for each function (be sure to use proper
notation):
5𝑥
a. 𝑓(𝑥) = |𝑥 − 2|
1 − 6𝑥
b. 𝑓(𝑥) = 1 + 2𝑥
3
c. 𝑓(𝑥) = (𝑥 − 2)3
5. Evaluate the limit using the substitution method.
a. lim
𝑥2+ 𝑥 + 2
𝑥+1
x→1
b. lim
√𝑥 + 1
𝑥−4
x→3
a. lim
x→1
𝑥−2
𝑥2 − 4
6. Use the graph below to evaluate the limits:
a. lim
d. lim
x → -3
b. lim
x→-∞
e. 𝑓(3)
x → -3
c. lim
f. When is the graph discontinuous?
x→∞
7. Using the graph below answer parts a-d.
a. 𝑓(2)
b. lim
x→2
c. lim
x→0
d. Name the intervals in which this function is continuous.
8. Using the graph below answer parts a-d.
a. lim
c. lim
x→0
x→0
b. lim
d. Is the function continuous when x = 1? Why?
x→1
e. Name the intervals in which this function is continuous.
f. When is the graph discontinuous?
9. Evaluate the limit.
a. lim
𝑥 3 − 4𝑥 2 + 7
3−6𝑥− 2𝑥 3
x→∞
b. lim
x→∞
𝑥2− 7
𝑥−4
c. lim
4− 𝑥 2
4𝑥 2 − 𝑥−2
x→∞
d. lim
x→∞
3𝑥 2 + 27
𝑥 3 − 27