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Introduction to calculus
Finding limits: properties of
How To
Given the limit of a function in quotient form, use factoring to evaluate it.
Factor the numerator and denominator completely.
Simplify by dividing any factors common to the numerator and denominator.
Evaluate the resulting limit, remembering to use the correct domain.
Evaluating the limit of a quotient by factoring
Evaluate
lim
x
→
2
(
x
2
−
6
x
+
8
x
−
2
)
.
Factor where possible, and simplify.
lim
x
→
2
(
x
2
−
6
x
+
8
x
−
2
)
=
lim
x
→
2
(
(
x
−
2
)
(
x
−
4
)
x
−
2
)
Factor the numerator
.
=
lim
x
→
2
(
(
x
−
2
)
(
x
−
4
)
x
−
2
)
Cancel the common factors
.
=
lim
x
→
2
(
x
−
4
)
Evaluate
.
=
2
−
4
=
−
2
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How To
Given a limit of a function containing a root, use a conjugate to evaluate.
If the quotient as given is not in indeterminate
(
0
0
)
form, evaluate directly.
Otherwise, rewrite the sum (or difference) of two quotients as a single quotient, using the
least common denominator (LCD) .
If the numerator includes a root, rationalize the numerator; multiply the numerator and denominator by the
conjugate of the numerator. Recall that
a
±
b
are conjugates.
Simplify.
Evaluate the resulting limit.
Evaluating a limit containing a root using a conjugate
Evaluate
lim
x
→
0
(
25
−
x
−
5
x
)
.
lim
x
→
0
(
25
−
x
−
5
x
)
=
lim
x
→
0
(
(
25
−
x
−
5
)
x
⋅
(
25
−
x
+
5
)
(
25
−
x
+
5
)
)
Multiply numerator and denominator by the conjugate
.
=
lim
x
→
0
(
(
25
−
x
)
−
25
x
(
25
−
x
+
5
)
)
Multiply:
(
25
−
x
−
5
)
⋅
(
25
−
x
+
5
)
=
(
25
−
x
)
−
25.
=
lim
x
→
0
(
−
x
x
(
25
−
x
+
5
)
)
Combine like terms
.
=
lim
x
→
0
(
−
x
x
(
25
−
x
+
5
)
)
Simplify
−
x
x
=
−
1.
=
−
1
25
−
0
+
5
Evaluate
.
=
−
1
5
+
5
=
−
1
10
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Evaluating the limit of a quotient of a function by factoring
Evaluate
lim
x
→
4
(
4
−
x
x
−
2
)
.
lim
x
→
4
(
4
−
x
x
−
2
)
=
lim
x
→
4
(
(
2
+
x
)
(
2
−
x
)
x
−
2
)
Factor.
=
lim
x
→
4
(
(
2
+
x
)
(
2
−
x
)
−
(
2
−
x
)
)
Factor
−1
out of the denominator
. Simplify
.
=
lim
x
→
4
−
(
2
+
x
)
Evaluate
.
=
−
(
2
+
4
)
=
−
4
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How To
Given a quotient with absolute values, evaluate its limit.
Try factoring or finding the LCD.
If the
limit cannot be found, choose several values close to and on either side of the input where the function is undefined.
Use the numeric evidence to estimate the limits on both sides.
Evaluating the limit of a quotient with absolute values
Evaluate
lim
x
→
7
|
x
−
7
|
x
−
7
.
The function is undefined at
x
=
7
, so we will try values close to 7 from the left and the right.
Left-hand limit:
|
6.9
−
7
|
6.9
−
7
=
|
6.99
−
7
|
6.99
−
7
=
|
6.999
−
7
|
6.999
−
7
=
−
1
Right-hand limit:
|
7.1
−
7
|
7.1
−
7
=
|
7.01
−
7
|
7.01
−
7
=
|
7.001
−
7
|
7.001
−
7
=
1
Since the left- and right-hand limits are not equal, there is no limit.
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Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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