Chapter 4 Limits and Continuity: Exercises

(Updated solution)

1. Given the graph of the function g(t) , find

1. lim− g(t) = [-1] 5. lim− g(t) = [2]

t→0 t→2

6. lim+ g(t) = [0]

2. lim+ g(t) = [-2] t→2
t→0
7. lim g(t) = [does not exist]
t→2
3. lim g(t) = [does not exist]
t→0 8. g(2) = [1]

4. g(0) = [-1] 9. lim g(t) = [3]

t→4

2. Use the graph to estimate the indicated function values and limits.

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 1. lim f (x) [-2] 5. lim f (x) [does not exist]
x→∞ x→−2

2. lim f (x) [ ∞ ] 6. lim+ f (x) [ ∞ ]

x→−∞ x→2

3. lim f (x) [ −∞ ] 7. lim− f (x) [ 0 ]

x→−2+ x→2

4. lim f (x) [ ∞ ] 8. lim f (x) [does not exist]

x→−2− x→2

3. Use the graph to estimate the indicated function values and limits.

1. (a) lim f (x) [-2] (d) f (−3) [1]

x→−3+

(b) lim f (x) [-2]

x→−3− (e) Is it possible to redefine f (−3) so
(c) lim f (x) [-2] that lim f (x) = f (−3) ? Explain.
x→−3 x→−3

2. (a) lim f (x) [3] (d) f (−2) [-3]

x→−2+

(b) lim f (x) [-3]

x→−2− (e) Is it possible to redefine f (−2) so
(c) lim f (x) [does not exist] that lim f (x) = f (−2) ? Explain.
x→−2 x→−2

3. (a) lim+ f (x) [2] (e) Is it possible to redefine f (0) so

x→0
that lim f (x) = f (0) ? Explain.
x→0
(b) lim− f (x) [2]
x→0

(c) lim f (x) [2]

x→0

(d) f (0) [does not exist]

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 4. (a) lim+ f (x) [-3] (d) f (−3) [3]
x→2

(b) lim− f (x) [3]

x→2 (e) Is it possible to redefine f (2) so
(c) lim f (x) [does not exist] that lim f (x) = f (2) ? Explain.
x→2 x→2

4. Find each limit.

1. lim (5x3 + 4) [139] |5x − 10|

x→3 11. lim [does not exist]
x→2 x − 2

2. lim (x4 − 2x + 3) [6] 12. lim (212 − 20x15 ) [- ∞ ]

x→−1
x→∞
4
3. lim [-2] 13. lim (−17x−1 ) [0]
x→−1 3x2 −5 x→−∞

x2 5x3 − 2x2 + 1 5
4. lim 2 [ 45 ] 14. lim [3]
x→−2 x + 1 x→−∞ 3x3 + 5
x2 − 6x + 9 7x2 + x + 11
5. lim [0] 15. lim [0]
x→3 x−3 x→−∞ 4 − x4
3x2 − x − 10 11 9x2 + x + 3
6. lim− [ 4 ] 16. lim [∞]
x→2 x2 − 4 x→−∞ 2−x
x4 − 1 17. lim e5x [ ∞ ]
7. lim+ [4] x→∞
x→1 x−1
18. lim e−x [ ∞ ]
x2 − 3x − 10 x→−∞
8. lim 2 [does not exist]
x→5 x − 10x + 25 1
19. lim 7 2 x [ ∞ ]
x→∞
|5x − 10|
9. lim− [-5]
x→2 x−2 20. lim ln 8 [ ln 8 ]
x→∞
|5x − 10|
10. lim+ [5] 21. lim 33x [0]
x→2 x−2 x→−∞


1 − x 2

if x ≤ 1,
5. Define f (x) = , find
1 + x 2

if x > 1

1. lim+ f (x) [1] 3. lim f (x) [1]

x→0 x→0

2. lim− f (x) [1] 4. f (0) [1]

x→0

|x − 1|
6. Define f (x) = , find
x−1

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 1. lim+ f (x) [1] 3. lim f (x) [does not exist]
x→1 x→1

2. lim− f (x) [-1] 4. f (1) [does not exist]

x→1

7. Use the graph to estimate the indicated function values and limits.

1. (a) lim− f (x) [2] (c) lim f (x) [does not exist]
x→1 x→1

(b) lim+ f (x) [1] (d) Is f continuous at x = 1 ? Why?

x→1

2. (a) lim− f (x) [2] (c) lim f (x) [2]

x→2 x→2

(b) lim+ f (x) [2] (d) Is f continuous at x = 2 ? Why?

x→2

3. (a) lim f (x) [0] (c) lim f (x) [0]

x→−1− x→−1

(b) lim f (x) [0] (d) Is f continuous at x = −1 ? Why?

x→−1+

4. (a) lim f (x) [1] (c) lim f (x) [1]

x→−2− x→−2

(b) lim f (x) [1] (d) Is f continuous at x = −2 ? Why?

x→−2+

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 f (2 + h) − f (2)
8. Compute lim when define
h→0 h

1. f (x) = 3x + 1 [3] 2. f (x) = x2 − 2 [4]

9. For the following functions f , locate all points of discontinuity, and discuss the
behavior of f at these points.

1 + x if x < 1,

1. f (x) = [x = 1]
5 − x if x ≥ 1



x2 if x < 1,

2. f (x) = [x = 1]
2x if x ≥ 1



x2 if x ≤ 2,

3. f (x) = [ f is continuous on (−∞, ∞) ]
2x if x > 2


10. For each function, determine where each function is continuous on (−∞, ∞) .

1. f (x) = 3x − 4 [ f is continuous at all points in (−∞, ∞) .]

x−2
2. g(x) = [All points in (−∞, ∞) are continuous except x = 3, −1 ]
(x − 3)(x + 1)
1 − x2
3. h(x) = [ f is continuous at all points in (−∞, ∞) .]
1 + x2
x2 + 4
4. j(x) = [All points in (−∞, ∞) are continuous except x = 4/5, −4/5 .]
4 − 25x2

x2 if x ≤ −3,






11. Let f (x) = 3a if − 3 < x ≤ 3,




b − 5x if x > 3.


1. When a = 2 and b = 6 , find the values of x where function f is discontinuous.

[ x = 3, −3 ]

2. Let a = 3 . Find b such that f is continuous for every x . [ b = 24 ]

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