NAME:
MATH 151
September 24, 2014
QUIZ 3
• Calculators are NOT allowed!
• Show all your work and indicate your final answer clearly. You will be graded not merely
on the final answer, but also on the work leading up to it.
1. (3 points) If 6x ≤ g(x) ≤ 3x4 − 3x2 + 6 for all x, evaluate lim g(x).
x→1
Solution: Since lim 6x = 6 and lim 3x4 − 3x2 + 6 = 6, the squeeze theorem gives that
x→1
x→1
lim g(x) = 6.
x→1
2. (3 points) Evaluate the following limit if it exists. If it does not exist, say so.
√
2− x
lim
x→4 4x − x2
.
Solution 1: Here we apply the identity a2 − b2 = (a + b)(a − b).
√
√
2− x
2− x
lim
= lim
x→4 4x − x2
x→4 x(4 − x)
√
2− x
√
√
= lim
x→4 x(2 −
x)(2 + x)
1
√
= lim
x→4 x(2 +
x)
1
=
16
NAME:
MATH 151
September 24, 2014
Solution 2: Here we multiply by the conjugate.
√
√
2− x
2− x
lim
= lim
x→4 4x − x2
x→4 x(4 − x)
√
√
2− x 2+ x
√
= lim
·
x→4 x(4 − x) 2 +
x
4−x
√
= lim
x→4 x(4 − x)(2 +
2)
1
√
= lim
x→4 x(2 +
x)
1
=
16
3.
( 3.5 points each) For the function g(x) defined at the right,
fill in the blanks or circle an answer:
a.
b.
lim g (x) = 4
x→1−
lim g (x) = 4
x→1+
c. g (1) = 3
d. lim g (x) = 4 or DOES NOT EXIST
x→1
e. g (x) is continuous from the left
f.
g (x) is continuous from the right
g. g (x) is continuous
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE

 3 + x2 if x < 1
3
if x = 1
g (x) =

4
if x > 1