MA161 - Quiz 5 - Fall 2015
Name:
Directions: Please show all your work leading to your answers. Having some correct work with an incorrect answer will
earn you partial credit.
1. Evaluate the limits, if they exist. If a limit does not exist, write DNE. Be sure to show your work. (3 points each)
(a) lim
x→3
x−3
x−4
x−3
3−3
Solution: lim
=
= 0.
x→3 x − 4
3−4
√
x−1
(b) lim
x→1 x − 1
Solution:
√
lim
x→1
√
√
x−1
x−1 x+1
√
= lim
x→1 x − 1
x−1
x+1
x−1
√
= lim
x→1 (x − 1)( x + 1)
1
= lim √
x→1
x+1
1
= .
2
x4 − 16
x→2 x2 + 2x − 8
Solution:
(c) lim
(x2 − 4)(x2 + 4)
x4 − 16
=
lim
x→2 (x + 4)(x − 2)
x→2 x2 + 2x − 8
(x + 2)(x − 2)(x2 + 4)
= lim
x→2
(x + 4)(x − 2)
(x + 2)(x2 + 4)
= lim
x→2
x+4
16
=
.
3
lim
(x + h)2 − x2
h→0
h
Solution:
(d) lim
(x + h)2 − x2
x2 + 2xh + h2 − x2
= lim
h→0
h→0
h
h
2xh + h2
= lim
h→0
h
= lim 2x + h
lim
h→0
= 2x.
1
2. Answer the following questions. (2 points each)
(a) Compute lim+ ln x. If the limit doesn’t exist, write DNE.
x→0
Solution: −∞ (from looking at the graph of y = ln x)
(b) What is the domain of ln x? Use this to explain why lim ln x doesn’t make sense.
x→0
Solution: The domain of ln x is (0, ∞). The limit doesn’t make sense because we can’t approach 0 from the left.
No numbers to the left of 0 are in the domain.
(c) What is the domain of ln x2 ?
Solution: All negative numbers are in the domain now since they get squared before being put into ln. The
domain is (−∞, 0) ∪ (0, ∞).
(d) Compute lim ln(x2 ). If the limit doesn’t exist, write DNE.
x→0
Solution: −∞
2