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Wize AP Calculus (AB) Textbook > Limits & Continuity

Computing Limits by Factoring

Indeterminate FormsPrevious Section
Computing Limits by Multiplying by the ConjugateNext 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}