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Limits by factoring Calculator

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1

Solved example of limits by factoring

$\lim_{x\to4}\left(\frac{x^2-16}{x^2+2x-24}\right)$
2

Factor the trinomial $x^2+2x-24$ finding two numbers that multiply to form $-24$ and added form $2$

$\begin{matrix}\left(-4\right)\left(6\right)=-24\\ \left(-4\right)+\left(6\right)=2\end{matrix}$
3

Thus

$\lim_{x\to4}\left(\frac{x^2-16}{\left(x-4\right)\left(x+6\right)}\right)$

Applying the power of a power property

$\lim_{x\to4}\left(\frac{\left(x+4\right)\left(x-4\right)}{\left(x-4\right)\left(x+6\right)}\right)$

Simplify the fraction by $x-4$

$\lim_{x\to4}\left(\frac{x+4}{x+6}\right)$
4

Factor the difference of squares $x^2-16$ as the product of two conjugated binomials

$\lim_{x\to4}\left(\frac{x+4}{x+6}\right)$
5

Evaluate the limit by replacing all occurrences of $x$ by $4$

$\frac{4+4}{4+6}$

Add the values $4$ and $4$

$\frac{8}{4+6}$

Add the values $4$ and $6$

$\frac{8}{10}$

Divide $8$ by $10$

$\frac{4}{5}$
6

Simplifying

$\frac{4}{5}$

Final Answer

$\frac{4}{5}$$\,\,\left(\approx 0.8\right)$

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