(3/19/08)
Section 1.2
More on finite limits
Overview:
In this section we find limits where the function is not defined at the limiting value of the
variable because it involves a denominator which is zero at that point. This is the most important type of
finite limit in calculus because derivatives are defined as such limits. We will find the limits by rewriting
the formulas for the functions.
Topics:
•
Limits of quotients of polynomials that tend to zero
•
Rationalizing differences of square roots
Limits of quotients of polynomials that tend to zero
At the beginning of the last section we considered a ball that falls
h
= 16
t
2
feet in
t
seconds (Figure 1) ,
so that the graph of the distance it falls after
t
= 0 is the parabola in Figure 2.
h
= 16
t
2
feet
Ball
Initial position
t
1
h
(feet)
16
50
h
= 16
t
2
16
t
2
t
(seconds)
FIGURE 1
FIGURE 2
We saw that because the ball falls 16
t
1

16 feet in the
t

1 seconds between time 1 to a later
time
t
, its average velocity in that time interval is
[Average velocity] =
16
t
2

16
t

1
feet per second
.
By calculating values of the average velocity from times
t
near 1, we predicted that the average velocity
would approach 32 feet per second as
t
approaches 1. We can now verify that prediction.
Example 1
What is the limit of lim
t
→
1
16
t
2

16
t

1
as
t
→
1?
Solution
We cannot apply Theorem 1 or Theorem 2 in Section 1.1 on limits of quotients of
functions and limits of rational functions because the denominator of
16
t
2

16
t

1
is zero
at
t
= 1. Instead, we factor the numerator to obtain
16
t
2

16
t

1
=
16(
t
2

1)
t

1
=
16(
t
+ 1)(
t

1)
t

1
.
14
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Section 1.2, More on finite limits
p. 15 (3/19/08)
Then we cancel the factor
t

1 in the numerator with the denominator to have for
t
negationslash
= 1,
16
t
2

16
t

1
= 16(
t
+ 1)
.
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 Summer '08
 Eggers
 Math, Limits, lim, Rational function, Square number

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