College Mathematics: Learning Worksheets Chapter 9
Name ________________________________ Date ______________ Class ____________
Goal: To find limits of functions
Section 9-1 Introduction to Limits
Definition: Limit
We write
lim ( )
xc
f
xL
or ()
f
xLas
x
c
if the functional value ()
f
xis close to the single real number L whenever x is close,
but not equal, to c (on either side of c).
Definition: One-sided limits
lim ( )
x
c
f
xL
is the limit of the function as x approaches the value c from the left.
lim ( )
x
c
f
xL
is the limit of the function as x approaches the value c from the right.
Properties of Limits:
2. lim
xc
x
c
f
4. lim[ () ()] lim () lim ()
xc xc xc
f
xgx fx gx
6. lim[ () ()] lim ()lim ()
f
xgx fx gx
7.
lim () lim ()
xc
xc
gx
8.
lim ( ) lim ( )
nn
xc xc
f
xfx
(the limit value must be positive for n even.)
College Mathematics: Learning Worksheets Chapter 9
230
In Problems 1–5, find each limit, if it exists
1.
2
lim (9 5)
x
x
→+(9(2) 5) 18 5 23=+=+=
2.
3
lim 5
x
x
5( 3) 15
5.
3
lim 8 1
x
x
→− −+ 8( 3) 1 24 1 25 5=−−+= += =
In Problems 6–8, find the value of the following limits given that
3
lim ( ) 5
x
fx
→= and
3
lim ( ) 4.
x
gx
→=−
f
7.
lim [3 ( ) 2 ( )]
f
xgx
8.
3
lim 3()
x
gx
3( 4) 12 6
===−
−−
College Mathematics: Learning Worksheets Chapter 9
231
9. Let
33if 1
() .
31if 1
xx
fx xx
⎧+<
⎪
=⎨+≥
⎪
⎩ Find:
a)
1
lim ( )
x
f
x
b)
1
lim ( )
x
f
x
x
c)
lim ( )
f
x
d) (1)
f
10. Let 56if 2
() .
24if 2
xx
fx xx
Find
a)
2
lim ( )
x
f
x
x
b)
2
lim ( )
x
f
x
x
College Mathematics: Learning Worksheets Chapter 9
232
c)
2
lim ( )
x
f
x
f
d) (2)
f
11. Let
287
() .
1
xx
fx x
⎛⎞
−+
=⎜⎟
⎜⎟
−
⎝⎠
Find
a)
1
lim ( )
x
f
x
f
22
8 7 (1) 8(1) 7 16
xx
−+ −−−+
f
College Mathematics: Learning Worksheets Chapter 9
12. Let 2
() .
2
x
fx x
Find
a)
2
lim ( )
x
f
x
b)
2
lim ( )
x
f
x
College Mathematics: Learning Worksheets Chapter 9
234
College Mathematics: Learning Worksheets Chapter 9
Name ________________________________ Date ______________ Class ____________
Goal: To find limits of functions as they approach infinity
Section 9-2 Infinite Limits and Limits at Infinity
Limits of Power Functions at Infinity:
If p is a positive real number and k is any real number except 0, then
1.
lim 0
p
x
k
x
2. lim 0
p
x
k
x
3. lim p
x
kx
4. lim p
x
kx
provided that
p
x
is a real number for negative values of x. The limits in 3 and 4
will be either positive or negative infinity, depending on k and p.
Limits of Rational Functions at Infinity:
There are three cases to consider:
2. If ,mnlim ( ) lim ( ) .
m
xx n
a
fx fx b
College Mathematics: Learning Worksheets Chapter 9
236
In Problems 1–3, find each limit. Use or when appropriate.
1. () 7
x
fx
x
=+
a)
7
lim ( )
x
f
x
−
→−
x
b)
7
lim ( )
x
f
x
+
→−
x
c)
lim ( )
f
x
2. 2
41
()
(5)
x
fx
x
a)
5
lim ( )
x
f
x
x
College Mathematics: Learning Worksheets Chapter 9
237
b)
5
lim ( )
x
f
x
x
f
3.
235
() 5
xx
fx x
+−
=−
a)
5
lim ( )
x
f
x
−
→
x
b)
5
lim ( )
x
f
x
+
→
x
c)
lim ( )
f
x
College Mathematics: Learning Worksheets Chapter 9
In Problems 4–6, find each function value and limit. Use or where appropriate.
4. 53
() 32
x
fx
x
−
=+
a) (20)f
b) (200)f
c) lim ( )
f
x
5
= because the function is a rational expression and ,mnthe
5. 2
5
()
322
x
fx
xx
a) (10)f
b) (100)f
c) lim ( )
f
x
College Mathematics: Learning Worksheets Chapter 9
239
6.
321
() 38
x
x
fx
x
a) ( 10)f
b) ( 100)f
c) lim ( )
f
x
In Problems 7–9, find the vertical and horizontal asymptotes for the following functions.
7. 8
() 5
x
fx
x
=+
8.
2
2
35
()
32
x
fx
x
−
=+
Vertical asymptotes are found by setting the denominator equal to zero. Since the
College Mathematics: Learning Worksheets Chapter 9
240
9.
2
2
45
()
815
xx
fx
xx
The function can be reduced as follows:
College Mathematics: Learning Worksheets Chapter 9
241
Name ________________________________ Date ______________ Class ____________
Goal: To determine if functions are continuous at specific points and intervals
Section 9-3 Continuity
Definition: Continuity
A function
f is continuous at the point
x
cif
1. lim ( )
xc
f
x
exists 2. ()
f
cexists 3. lim ( ) ( )
xc
f
xfc
Continuity Properties:
1. A constant function ( ) ,
f
xkwhere k is a constant, is continuous for all x.
3. A polynomial function is continuous for all x.
5. For n an odd positive integer greater than 1, ()
n
f
xis continuous wherever
()
f
xis continuous.
6. For n an even positive integer, ()
n
f
xis continuous wherever ()
f
xis
continuous and nonnegative.
Constructing Sign Charts:
2. Plot the numbers found in step 1 on a real-number line, dividing the number
line into intervals.
f
f
4. Construct a sign chart, using the real-number line in step 2.
College Mathematics: Learning Worksheets Chapter 9
242
In Problems 1–5, using the continuity properties, determine where each of the functions are
continuous.
1. 32
() 5 3 2 8
f
xxx x=−+−
2.
32
2
437
()
936
xxx
fx
xx
++−
=−−
3. 2
32
()
3
x
fx
x
4. 3
() 2 5fx x=−
5. 2
() 25fx x
243
6. 2
42970xx
Find the partition numbers:
(0) 4(0) 29(0) 7
f
2
(1) 4(1) 29(1) 7
f
2
(8) 4(8) 29(8) 7
f
College Mathematics: Learning Worksheets Chapter 9
244
7.
32
30
5
xx
x
Find the partition numbers:
32
30
xx
50x
(6) 3(6) 108
(4) 3(4) 16
(1) 3(1) 2
(1) 3(1) 4
College Mathematics: Learning Worksheets Chapter 9
Name ________________________________ Date ______________ Class ____________
Goal: To find the first derivative of a function using the four-step process.
Section 9-4 The Derivative
Definition: Average Rate of Change
For
(),yfxthe average rate of change from
x
a to
x
ahis
where h is the distance from the initial value of x to the final value of x.
Definition: Instantaneous Rate of Change
For
(),yfxthe instantaneous rate of change at
x
ais
f
This formula is also used to find the slope of a graph at the point ( , ( ))afa and to
find the first derivative of a function, ( ).
f
x
Procedure: Finding the first derivative:
1. Find ( ).
f
xh
f
3. Find
.
f
h
4. Find
0
()()
lim .
h
f
xh fx
h
College Mathematics: Learning Worksheets Chapter 9
246
In Problems 1–3, use the four step procedure to find ( )
f
x
and then find (1),f(2),f and
(3).f
1. ( ) 9 4
f
xx=+
Step 1: ( ) 9( ) 4 9 9 4fx h x h x h+= ++=++
f
2. 2
() 2 4 7fx x x=− + −
Step 1: 2
22
22
()2()4()7
2( 2 ) 4( ) 7
()242447
fx h x h x h
xxhh xh
fx h x xh h x h
+=− + + +−
=− + + + + −
+=− − − ++−
College Mathematics: Learning Worksheets Chapter 9
3. 2
() 5
x
fx x
Step 1: 2( ) 2 2
()
()5 5
x
hxh
fx h xh xh
Step 2: 22 2
()() 55
xh x
fx h fx xh x
(5)
x
(1 5) ( 4)
(2 5) ( 3)
(3 5) ( 2)
248
4. The profit, in hundreds of dollars, from the sale of x items is given by
2
() 5 3.Px x x=+−
a) Find the average rate of change of profit from x = 2 to x = 4.
b) Find the instantaneous rate of change equation using the four–step procedure.
c) Using the equation found in part b, find the instantaneous rate of change when
x = 2 and interpret the results.
Solution:
a) First find the function values at 2 and 4. Note that 42 2.h
Now use the average rate of change formula:
b) Step 1: 2
22
()()5()3
(2 )5()3
Px h x h x h
xxhh xh
+=+ + +−
=++++−