Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 7
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411. Comprehension check:
(a) Define what it means for f (x) to be continuous at the point x = a. What does it mean if f (x) is
continuous on the interval [a, b]? What does it mean to say f (x) is continuous?
(b) There are three distinct ways in which a function will fail to be continuous at a point x = a.
Describe the three types of discontinuity. Provide a sketch and an example of each type.
(c) True or false? Every function is continuous on its domain.
(d) True or false? The sum, difference, and product of continuous functions are all continuous.
(e) If f (x) is continuous at x = a, what can you say about lim
f (x)?
x
a
2. Using the definition of continuity and properties of limits, show that the following functions are con-
tinuous at the given point a.
(a) f (x) = π, a = 1
2
x
+ 3x + 1
(b) f (x) =
, a =
1
x + 3
2
(c) f (x) =
x
9, a = 4
3. Give the intervals of continuity for the following functions.
x + 1
2
x
+ 1
if x
0
(a) f (x) =
2
x
+ 4x + 3
(d) f (x) =
x + 1
if 0 < x < 2
x
(b) f (x) =
2
(x
2)
if x
2
2
x
+ 1
2
(c) f (x) =
2x
3 + x
4. If lim
(hf (h)) = 15, can you find lim
f (h)? Use the limit laws to justify your answer.
h
3
h
3
2
cx
5 if x < 1
10
if x = 1
5. Let c be a number and consider the function f (x) =
.
1
2c
if x > 1
x
(a) Find all numbers c such that lim
f (x) exists.
x
1
(b) Is there a number c such that f (x) is continuous at x = 1? Justify your answer.
2
2x
+ 3x
if x
4
6. Find parameters a and b so that f (x) =
ax + b
if
4 < x < 3
is continuous.
3
2
x
+ 4x
5 if 3
x
7. Suppose that f (x) and g(x) are continuous functions where f (2) = 5 and g(6) = 1. Compute the
following:
2
[f (x)]
+ x
(a) lim
.
3x + 2
x
2
g(x) + 4x
(b) lim
x
f
g(x)
x
6
3
x 6
for x = 6,
x 6
8. Suppose that: f (x) =
1
for x = 6
Determine the points at which the function f (x) is discontinuous and state the type of discontinuity.
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