© Dr. Rosanna Pearlstein, Spring 2015
Week In Review III- Sections 2.2, 2.3, 2.5
1.
Given the graph of 𝑓(𝑥) below, complete items a-h below.
a. lim𝑥→−6− 𝑓(𝑥) =
b. lim𝑥→−3− 𝑓(𝑥) =
c. lim𝑥→−3+ 𝑓(𝑥) =
d. lim𝑥→−3 𝑓(𝑥) =
e. lim𝑥→1 𝑓(𝑥) =
f. lim𝑥→4 𝑓(𝑥) =
g. lim𝑥→7 𝑓(𝑥) =
h. Find all vertical asymptotes.
2. For the function 𝑔(𝑥) defined below, find the following limits if they exist, and answer “DNE” if they don’t
exist.
a. lim𝑥→0 𝑔(𝑥)
b. lim𝑥→3 𝑔(𝑥)
𝑥<0
𝑥2
0
≤
𝑥<3
𝑔(𝑥) = { 1
𝑥
=3
−2
𝑥>3
4−𝑥
𝑥 2 −3𝑥+2
3. For the function 𝑓(𝑥) = (𝑥−2)(𝑥2 +2𝑥−15)
a. Is there any point 𝑎 where lim𝑥→𝑎 𝑓(𝑥) does not exist?
b. Find all of the vertical asymptotes of the graph of f and determine the behavior of the function near the
asymptotes.
4. Find the limits:
a. lim𝑥→−5+
(𝑥−2)
(𝑥+5)
b. lim𝑥→−5+
(𝑥−2)4
(𝑥+5)
c. lim𝑥→−5−
𝑥(𝑥−2)
(𝑥+5)
© Dr. Rosanna Pearlstein, Spring 2015
Section 2.3
5. Evaluate each of the following limits, or explain why the limit does not exist, by evaluating the right and left
hand limits. Comment on vertical asymptotes when appropriate.
a. lim𝑥→7
2𝑥
(2𝑥−7)
b. lim𝑥→0
𝑥 2 +4𝑥
𝑥
c. limℎ→0
√4+ℎ−2
ℎ
3𝑥 1
−
𝑥+2 5
d. lim𝑥→1
14𝑥−2
e. lim𝑥→3
|𝑥−3|
𝑥−3
7
f. lim𝑥→5 √𝑥 − 5
g. lim𝑥→−3 √𝑥 2
h. lim𝑥→0
sin(2𝑥)
sin(3𝑥)
i. lim𝑥→3
√4−𝑥 − 1
√19−𝑥 − 4
𝑥−2
𝑥3 − 8
i.
lim𝑥→2
j.
lim𝑥→0 (𝑥 2 ) (𝑐𝑜𝑠 ( 3 ))
1
𝑥
k. lim𝑥→0 √𝑥 𝑠𝑖𝑛4 𝑥
6. Investigate the limit of the following function as x approaches 2 and sketch the graph of 𝑦 = 𝑓(𝑥).
𝑓(𝑥) =
𝑥2 − 4
|𝑥 − 2|
© Dr. Rosanna Pearlstein, Spring 2015
Section 2.5
7. Explain why the function below is discontinuous at a=4. Could something be changed that would make g everywhere
continuous? Is the line x = 4 a vertical asymptote for the graph of g?
𝑥 2 − 2𝑥 − 8
𝑔(𝑥) = {
𝑥−4
3
𝑥≠4
𝑥=4
8. Thanks to archives.math.utk.edu. Given the piecewise-defined function below, find the value(s) of c which make the
function continuous.
2 𝑥 >4
𝑓(𝑥) = {(𝑥 − 2𝑐)
𝑥≤4
2𝑥 − 𝑐
5−√𝑥
9. Can the discontinuity of ℎ(𝑥) = 25−𝑥 at 𝑥 = 25 be removed? Explain why yes or why no. Does the graph of the
function have any vertical asymptotes?
10. Show that there is a number α in [1, 4] such that 𝛼 2 − √𝛼 = 9.