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Wize University Calculus 1 Textbook > Limits

Computing Limits by Factoring

Indeterminate FormsPrevious Section
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Computing Limits by Factoring

One common way of dealing with the 00\frac{0}{0} indeterminate form is to factor the numerator or the denominator (or both) and cancel things out.

Wize Concept
Following identities can be useful in extracting a problematic factor:
a2−b2=(a−b)(a+b)a3−b3=(a−b)(a2+ab+b2)a3+b3=(a+b)(a2−ab+b2)\begin{array}{ll}a^2-b^2=(a-b)(a+b)\\a^3-b^3=(a-b)(a^2+ab+b^2)\\a^3+b^3=(a+b)(a^2-ab+b^2)\end{array}


Note: You can always use long division to break down any complicated polynomial into its factors!
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Example: Computing Limits by Factoring

Find the following limit

lim⁡x→5x2−25x−5\displaystyle\lim_{x\rightarrow5}\frac{x^2-25}{x-5}

lim⁡x→5x2−25x−5=lim⁡x→5(x−5)(x+5)x−5=lim⁡x→5x+5=10\lim_{x \to 5}\dfrac{x^2-25}{x-5}\\ \text{} \\=\lim_{x \to 5}\dfrac{\cancel{(x-5)}(x+5)}{\cancel{x-5}}\\ \text{} \\=\lim_{x \to 5}x+5\\ \text{} \\=10

Find the following limit

lim⁡x→3x−3x3−27\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}


Extra Practice
Limits: Factoring Terms
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
Limits: Factoring Terms
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
Limits: Factoring
Evaluate
lim⁡x→27x2/3−9x1/3−3\lim\limits_{x\to 27} \frac{x^{2/3}-9}{x^{1/3}-3}
Limits: Factoring
Evaluate
lim⁡x→01−cos⁡2xsin⁡22x\lim\limits_{x\to 0} \frac{1-\cos 2x}{\sin^22x}
Limits: Factoring Terms
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
Limit Techniques: Factoring
lim⁡x→3x−3x3−27\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
Limit Techniques: Factoring
lim⁡x→3x−3x3−27\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
Limit Techniques: Factoring
lim⁡x→3x−3x3−27\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
One-sided limits: Evaluating limits by factoring
Practice: Evaluating limits
Evaluate lim⁡x→2−x2−3x+2∣x−2∣\lim_{x\rightarrow2^-}\frac{x^2-3x+2}{\left|x-2\right|}
The limit lim⁡x→2x2−5x+6x2−4\displaystyle\lim_{x\to2}\frac{x^2-5x+6}{x^2-4} evaluates to:
Find the following limit: lim⁡x→5x2−25x−5\displaystyle\lim_{x\rightarrow5}\frac{x^2-25}{x-5}
Limits: Factoring
Evaluate lim⁡x→01−cos⁡2xsin⁡22x\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos{2x}}{\sin^2{2x}}.
Limit with Fractional Powers
Evaluate lim⁡x→27x2/3−9x1/3−3\displaystyle \lim_{x\rightarrow 27}\frac{x^{2/3}-9}{x^{1/3}-3}.
Limits: Cubic
Find the limit lim⁡x→1x3−1x−1\displaystyle\lim\limits_{x\to 1}\frac{x^{3}-1}{x-1}
Limits: Factoring Terms
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
Limits: Factoring Terms
lim⁡x→2x3−5x2+11x−10x−2\displaystyle\lim_{x\rightarrow2}\frac{x^3-5x^2+11x-10}{x-2}
lim⁡x→1x3−14x2+7x+6x−1\displaystyle\lim_{x\rightarrow1}\frac{x^3-14x^2+7x+6}{x-1}
Limits: Factoring Terms
lim⁡x→1x9−1x−1\displaystyle\lim_{x\rightarrow1}\frac{x^9-1}{x-1}
Limits: Factoring Terms
lim⁡x→3x4−81x−3\displaystyle\lim_{x\rightarrow3}\frac{x^4-81}{x-3}
lim⁡x→3x4−81x−3\displaystyle\lim_{x\rightarrow3}\frac{x^4-81}{x-3}
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
Practice: Factorable Indeterminate Forms
Cancelling Common Factors: Evaluate lim⁡x→2x2+x−6x−2\lim_{x\to 2}\limits\dfrac{x^2 + x - 6}{x - 2}
Limits: Factoring
Evaluate
lim⁡x→27x2/3−9x1/3−3\lim\limits_{x\to 27} \frac{x^{2/3}-9}{x^{1/3}-3}
Limits: Factoring
Evaluate
lim⁡x→01−cos⁡2xsin⁡22x\lim\limits_{x\to 0} \frac{1-\cos 2x}{\sin^22x}
Limits: Factoring
Evaluate lim⁡x→01−cos⁡2xsin⁡22x\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos{2x}}{\sin^2{2x}}.
Limit with Fractional Powers
Evaluate lim⁡x→27x2/3−9x1/3−3\displaystyle \lim_{x\rightarrow 27}\frac{x^{2/3}-9}{x^{1/3}-3}.
Limits: Cubic
Find the limit lim⁡x→1x3−1x−1\displaystyle\lim\limits_{x\to 1}\frac{x^{3}-1}{x-1}
Find the following limit: lim⁡x→5x2−25x−5\displaystyle\lim_{x\rightarrow5}\frac{x^2-25}{x-5}
The limit lim⁡x→2x2−5x+6x2−4\displaystyle\lim_{x\to2}\frac{x^2-5x+6}{x^2-4} evaluates to:
The value of the limit lim⁡x→1x−1x2−3x+2\lim_{x\to1}\frac{\sqrt{x}-1}{x^2-3x+2} is
Limits: Factoring (Common Denominator)
Solve the limit lim⁡x→5x−x+10x−225−x2\displaystyle \lim\limits_{x\to 5}\frac{x-\frac{x+10}{x-2}}{25-x^2}.
Limits: Factoring
What is the limit lim⁡x→0sin⁡2xcos⁡x1−cos⁡x\lim\limits_{x\to0}\frac{\sin^2 x\cos x}{1-\cos x}?
The limit lim⁡x→2x2−5x+6x2−4\lim\limits_{x\rightarrow 2}\frac{x^2-5x+6}{x^2-4} evaluates to
Limits: Factoring terms
Evaluate the following limit
lim⁡x→4x(x−2)x−6x+8\displaystyle \lim_{x\rightarrow 4}\frac{x(\sqrt{x}-2)}{x-6\sqrt{x}+8}
Evaluate lim⁡x→2−x2−3x+2∣x−2∣\lim_{x\rightarrow2^-}\frac{x^2-3x+2}{\left|x-2\right|}.
🦊 TRICKY! Determine the value of lim⁡x→−1x3+1x+1\displaystyle \lim_{x\rightarrow -1}\frac{x^3+1}{x+1}.
🦊 TRICKY! Determine the value of lim⁡x→2x3−8x−2\displaystyle \lim_{x \to 2} \frac{x^3-8}{x-2}.
Evaluate lim⁡x→∞4x2+1−2x\displaystyle \lim_{x\rightarrow \infty}\sqrt{4x^2+1}-2x.
Evaluate the limit, lim⁡x→3x2−9x−3\displaystyle\lim_{x\rightarrow3}\frac{x^2-9}{x-3}.
Evaluate the limit, lim⁡x→2x+7−3x−2\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x+7}-3}{x-2}.
Evaluate the limit, lim⁡x→0x2+x−2−x\displaystyle \lim_{x\rightarrow0}\frac{x}{\sqrt{2+x}-\sqrt{2-x}}.
Find the following limit: lim⁡x→3x2−9x−3\displaystyle \lim_{x\rightarrow3}\frac{x^2-9}{\sqrt{x}-\sqrt{3}}.
Limits: Factoring
Evaluate the limit lim⁡x→8x3−2x2−64\displaystyle \lim\limits_{x\to 8}\frac{\sqrt[3]{x}-2}{x^2-64}.
One-sided limits: Evaluating limits by factoring
Practice: Evaluating limits
Evaluate lim⁡x→2−x2−3x+2∣x−2∣\lim_{x\rightarrow2^-}\frac{x^2-3x+2}{\left|x-2\right|}
Limits by Factoring Terms
Practice: 0/0 Limit
lim⁡h→0 4−(5h+2)2h=\displaystyle\lim_{h\to0}\ \frac{4-\left(5h+2\right)^2}{h}=
Evaluating Limits: Factoring terms
Evaluate the following limit, or state that it does not exist and why.
lim⁡x→1x3−2x2−5x+6x2−1\lim_{x \rightarrow 1} \frac{x^3 - 2x^2 - 5x + 6}{x^2 - 1}
Limit Techniques: Factoring
lim⁡x→3x−3x3−27\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}