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Limit Techniques: Factoring
Related Topics
Wize University Calculus 1 Textbook > Limits
Computing Limits by Factoring
3 Activities
lim
x
→
3
x
−
3
x
3
−
27
\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
x
→
3
lim
x
3
−
27
x
−
3
1
3
\frac13
3
1
1
1
1
1
27
\frac1{27}
27
1
Does not exist
I don't know
Check Submission
More Computing Limits by Factoring Questions:
Limits: Factoring Terms
lim
x
→
1
x
5
−
1
x
−
1
\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
x
→
1
lim
x
−
1
x
5
−
1
Limits: Factoring Terms
lim
x
→
1
x
5
−
1
x
−
1
\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
x
→
1
lim
x
−
1
x
5
−
1
Limits: Factoring
Evaluate
lim
x
→
27
x
2
/
3
−
9
x
1
/
3
−
3
\lim\limits_{x\to 27} \frac{x^{2/3}-9}{x^{1/3}-3}
x
→
27
lim
x
1/3
−
3
x
2/3
−
9
Limits: Factoring
Evaluate
lim
x
→
0
1
−
cos
2
x
sin
2
2
x
\lim\limits_{x\to 0} \frac{1-\cos 2x}{\sin^22x}
x
→
0
lim
sin
2
2
x
1
−
cos
2
x
Limits: Factoring Terms
lim
x
→
1
x
5
−
1
x
−
1
\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
x
→
1
lim
x
−
1
x
5
−
1
Limit Techniques: Factoring
lim
x
→
3
x
−
3
x
3
−
27
\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
x
→
3
lim
x
3
−
27
x
−
3
Limit Techniques: Factoring
lim
x
→
3
x
−
3
x
3
−
27
\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
x
→
3
lim
x
3
−
27
x
−
3
One-sided limits: Evaluating limits by factoring
Practice: Evaluating limits
Evaluate
lim
x
→
2
−
x
2
−
3
x
+
2
∣
x
−
2
∣
\lim_{x\rightarrow2^-}\frac{x^2-3x+2}{\left|x-2\right|}
lim
x
→
2
−
∣
x
−
2
∣
x
2
−
3
x
+
2
The limit
lim
x
→
2
x
2
−
5
x
+
6
x
2
−
4
\displaystyle\lim_{x\to2}\frac{x^2-5x+6}{x^2-4}
x
→
2
lim
x
2
−
4
x
2
−
5
x
+
6
evaluates to:
Find the following limit:
lim
x
→
5
x
2
−
25
x
−
5
\displaystyle\lim_{x\rightarrow5}\frac{x^2-25}{x-5}
x
→
5
lim
x
−
5
x
2
−
25
Limits: Factoring
Evaluate
lim
x
→
0
1
−
cos
2
x
sin
2
2
x
\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos{2x}}{\sin^2{2x}}
x
→
0
lim
sin
2
2
x
1
−
cos
2
x
.
Limit with Fractional Powers
Evaluate
lim
x
→
27
x
2
/
3
−
9
x
1
/
3
−
3
\displaystyle \lim_{x\rightarrow 27}\frac{x^{2/3}-9}{x^{1/3}-3}
x
→
27
lim
x
1/3
−
3
x
2/3
−
9
.
Limits: Cubic
Find the limit
lim
x
→
1
x
3
−
1
x
−
1
\displaystyle\lim\limits_{x\to 1}\frac{x^{3}-1}{x-1}
x
→
1
lim
x
−
1
x
3
−
1
Limits: Factoring Terms
lim
x
→
1
x
5
−
1
x
−
1
\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
x
→
1
lim
x
−
1
x
5
−
1
Limits: Factoring Terms
lim
x
→
2
x
3
−
5
x
2
+
11
x
−
10
x
−
2
\displaystyle\lim_{x\rightarrow2}\frac{x^3-5x^2+11x-10}{x-2}
x
→
2
lim
x
−
2
x
3
−
5
x
2
+
11
x
−
10
lim
x
→
1
x
3
−
14
x
2
+
7
x
+
6
x
−
1
\displaystyle\lim_{x\rightarrow1}\frac{x^3-14x^2+7x+6}{x-1}
x
→
1
lim
x
−
1
x
3
−
14
x
2
+
7
x
+
6
Limits: Factoring Terms
lim
x
→
1
x
9
−
1
x
−
1
\displaystyle\lim_{x\rightarrow1}\frac{x^9-1}{x-1}
x
→
1
lim
x
−
1
x
9
−
1
Limits: Factoring Terms
lim
x
→
3
x
4
−
81
x
−
3
\displaystyle\lim_{x\rightarrow3}\frac{x^4-81}{x-3}
x
→
3
lim
x
−
3
x
4
−
81
lim
x
→
3
x
4
−
81
x
−
3
\displaystyle\lim_{x\rightarrow3}\frac{x^4-81}{x-3}
x
→
3
lim
x
−
3
x
4
−
81
lim
x
→
1
x
5
−
1
x
−
1
\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
x
→
1
lim
x
−
1
x
5
−
1
Practice: Factorable Indeterminate Forms
Cancelling Common Factors: Evaluate
lim
x
→
2
x
2
+
x
−
6
x
−
2
\lim_{x\to 2}\limits\dfrac{x^2 + x - 6}{x - 2}
x
→
2
lim
x
−
2
x
2
+
x
−
6
Limits: Factoring
Evaluate
lim
x
→
27
x
2
/
3
−
9
x
1
/
3
−
3
\lim\limits_{x\to 27} \frac{x^{2/3}-9}{x^{1/3}-3}
x
→
27
lim
x
1/3
−
3
x
2/3
−
9
Limits: Factoring
Evaluate
lim
x
→
0
1
−
cos
2
x
sin
2
2
x
\lim\limits_{x\to 0} \frac{1-\cos 2x}{\sin^22x}
x
→
0
lim
sin
2
2
x
1
−
cos
2
x
Limits: Factoring
Evaluate
lim
x
→
0
1
−
cos
2
x
sin
2
2
x
\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos{2x}}{\sin^2{2x}}
x
→
0
lim
sin
2
2
x
1
−
cos
2
x
.
Limit with Fractional Powers
Evaluate
lim
x
→
27
x
2
/
3
−
9
x
1
/
3
−
3
\displaystyle \lim_{x\rightarrow 27}\frac{x^{2/3}-9}{x^{1/3}-3}
x
→
27
lim
x
1/3
−
3
x
2/3
−
9
.
Limits: Cubic
Find the limit
lim
x
→
1
x
3
−
1
x
−
1
\displaystyle\lim\limits_{x\to 1}\frac{x^{3}-1}{x-1}
x
→
1
lim
x
−
1
x
3
−
1
Find the following limit:
lim
x
→
5
x
2
−
25
x
−
5
\displaystyle\lim_{x\rightarrow5}\frac{x^2-25}{x-5}
x
→
5
lim
x
−
5
x
2
−
25
The limit
lim
x
→
2
x
2
−
5
x
+
6
x
2
−
4
\displaystyle\lim_{x\to2}\frac{x^2-5x+6}{x^2-4}
x
→
2
lim
x
2
−
4
x
2
−
5
x
+
6
evaluates to:
The value of the limit
lim
x
→
1
x
−
1
x
2
−
3
x
+
2
\lim_{x\to1}\frac{\sqrt{x}-1}{x^2-3x+2}
lim
x
→
1
x
2
−
3
x
+
2
x
−
1
is
Limits: Factoring (Common Denominator)
Solve the limit
lim
x
→
5
x
−
x
+
10
x
−
2
25
−
x
2
\displaystyle \lim\limits_{x\to 5}\frac{x-\frac{x+10}{x-2}}{25-x^2}
x
→
5
lim
25
−
x
2
x
−
x
−
2
x
+
10
.
Limits: Factoring
What is the limit
lim
x
→
0
sin
2
x
cos
x
1
−
cos
x
\lim\limits_{x\to0}\frac{\sin^2 x\cos x}{1-\cos x}
x
→
0
lim
1
−
c
o
s
x
s
i
n
2
x
c
o
s
x
?
The limit
lim
x
→
2
x
2
−
5
x
+
6
x
2
−
4
\lim\limits_{x\rightarrow 2}\frac{x^2-5x+6}{x^2-4}
x
→
2
lim
x
2
−
4
x
2
−
5
x
+
6
evaluates to
Limits: Factoring terms
Evaluate the following limit
lim
x
→
4
x
(
x
−
2
)
x
−
6
x
+
8
\displaystyle \lim_{x\rightarrow 4}\frac{x(\sqrt{x}-2)}{x-6\sqrt{x}+8}
x
→
4
lim
x
−
6
x
+
8
x
(
x
−
2
)
Evaluate
lim
x
→
2
−
x
2
−
3
x
+
2
∣
x
−
2
∣
\lim_{x\rightarrow2^-}\frac{x^2-3x+2}{\left|x-2\right|}
lim
x
→
2
−
∣
x
−
2
∣
x
2
−
3
x
+
2
.
🦊
TRICKY!
Determine the value of
lim
x
→
−
1
x
3
+
1
x
+
1
\displaystyle \lim_{x\rightarrow -1}\frac{x^3+1}{x+1}
x
→
−
1
lim
x
+
1
x
3
+
1
.
🦊
TRICKY!
Determine the value of
lim
x
→
2
x
3
−
8
x
−
2
\displaystyle \lim_{x \to 2} \frac{x^3-8}{x-2}
x
→
2
lim
x
−
2
x
3
−
8
.
Evaluate
lim
x
→
∞
4
x
2
+
1
−
2
x
\displaystyle \lim_{x\rightarrow \infty}\sqrt{4x^2+1}-2x
x
→
∞
lim
4
x
2
+
1
−
2
x
.
Evaluate the limit,
lim
x
→
3
x
2
−
9
x
−
3
\displaystyle\lim_{x\rightarrow3}\frac{x^2-9}{x-3}
x
→
3
lim
x
−
3
x
2
−
9
.
Evaluate the limit,
lim
x
→
2
x
+
7
−
3
x
−
2
\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x+7}-3}{x-2}
x
→
2
lim
x
−
2
x
+
7
−
3
.
Evaluate the limit,
lim
x
→
0
x
2
+
x
−
2
−
x
\displaystyle \lim_{x\rightarrow0}\frac{x}{\sqrt{2+x}-\sqrt{2-x}}
x
→
0
lim
2
+
x
−
2
−
x
x
.
Find the following limit:
lim
x
→
3
x
2
−
9
x
−
3
\displaystyle \lim_{x\rightarrow3}\frac{x^2-9}{\sqrt{x}-\sqrt{3}}
x
→
3
lim
x
−
3
x
2
−
9
.
Limits: Factoring
Evaluate the limit
lim
x
→
8
x
3
−
2
x
2
−
64
\displaystyle \lim\limits_{x\to 8}\frac{\sqrt[3]{x}-2}{x^2-64}
x
→
8
lim
x
2
−
64
3
x
−
2
.
One-sided limits: Evaluating limits by factoring
Practice: Evaluating limits
Evaluate
lim
x
→
2
−
x
2
−
3
x
+
2
∣
x
−
2
∣
\lim_{x\rightarrow2^-}\frac{x^2-3x+2}{\left|x-2\right|}
lim
x
→
2
−
∣
x
−
2
∣
x
2
−
3
x
+
2
Limits by Factoring Terms
Practice: 0/0 Limit
lim
h
→
0
4
−
(
5
h
+
2
)
2
h
=
\displaystyle\lim_{h\to0}\ \frac{4-\left(5h+2\right)^2}{h}=
h
→
0
lim
h
4
−
(
5
h
+
2
)
2
=
Evaluating Limits: Factoring terms
Evaluate the following limit, or state that it does not exist and why.
lim
x
→
1
x
3
−
2
x
2
−
5
x
+
6
x
2
−
1
\lim_{x \rightarrow 1} \frac{x^3 - 2x^2 - 5x + 6}{x^2 - 1}
x
→
1
lim
x
2
−
1
x
3
−
2
x
2
−
5
x
+
6
Limit Techniques: Factoring
lim
x
→
3
x
−
3
x
3
−
27
\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
x
→
3
lim
x
3
−
27
x
−
3
lim
x
→
1
x
5
−
1
x
−
1
\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
x
→
1
lim
x
−
1
x
5
−
1