MATH 151 Engineering Mathematics I
Week In Review
Fall, 2015, Problem Set 3 (2.2, 2.3, 2.5)
JoungDong Kim
1. State the value of the given quantity, if it exists, from the given graph.
a. lim g(x)
b. lim g(x)
c. lim+ g(x)
d. lim g(x)
e. g(2)
f. lim g(x)
g.
h. lim g(x)
x→1
x→2
lim g(x)
x→−1−
x→0
x→0
x→−2
x→−1
1
2. Find the limit.
(a) lim+
6
x−5
(b) lim−
6
x−5
x→5
x→5
6
x→5 x − 5
(c) lim
1
x→3 (x − 3)8
(d) lim
(e) lim
x−1
+ 2)
x→0 x2 (x
(f)
lim +
x→−2
x−1
+ 2)
x2 (x
2
3. Evaluate the limit.
(a) lim (x2 + x + 1)5
x→−2
√
√
(b) lim ( 3 x + 3 x)
x→64
x2 − x − 12
x→−3
x+3
(c) lim
x2 − x − 2
x→1
x+1
(d) lim
(e) lim
√
t→0
2−t−
t
√
2
(3 + h)−1 − 3−1
(f) lim
h→0
h
t2 − t
(g) lim 2t − 3,
t→1
t−1
3
4. If 3x ≤ f (x) ≤ x3 + 2 for 0 ≤ x ≤ 2, evaluate lim f (x).
x→1
5. Find the limit.
(a)
lim −
x→−4
|x + 4|
x+4
2x2 − 3x
x→1.5 |2x − 3|
(b) lim
|x − 2|
x→2 x − 2
(c) lim
4


x
6. Let f (x) = x2


8−x
if x < 0
if 0 < x ≤ 2
if x > 2
evaluate each of the following limits if it exists.
(a) lim+ f (x)
x→0
(b) lim f (x)
x→0
(c) lim f (x)
x→1
(d) lim− f (x)
x→2
(e) lim+ f (x)
x→2
(f) lim f (x)
x→2
5
7. Use the definition of continuity and the properties of limit to show that the function f (x) =
x4 − 5x3 + 6 is continuous at a = 3.
8. Explain why the following function is not continous everywhere.
 2
 x − 2x − 8
if x 6= 4
f (x) =
x−4

3
if x = 4
6
9. Find the values of c and d that make f continous on R.


if x < 1
2x
2
f (x) = cx + d
if 1 ≤ x ≤ 2


4x
ifx > 2
√
3− x
is given. If the discontinuity at x = 9 is removable, find a function g that
10. Let f (x) =
9−x
agrees with f for x 6= 9 and is continuous on R.
7
11. Use the Intermediate Value Theorem to show that there is a root of the given equation in the
given interval.
x3 + 2x = x2 + 1, (0, 1)
8