Differential and Integral Calculus by Feliciano and Uy, Exercise 1.3, Problem 2

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PROBLEM:

Evaluate lim⁡x→2(x2+2x−83x−6)\displaystyle \lim\limits_{x\to 2}\left(\frac{x^2+2x-8}{3x-6}\right)

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SOLUTION:

A straight substitution of x=2x=2 leads to the indeterminate form 00\frac{0}{0}  which is meaningless.

Therefore, to evaluate the limit of the given function, we proceed as follows

lim⁡x→2(x2+2x−83x−6)=lim⁡x→2((x+4)(x−2)3(x−2))=lim⁡x→2(x+43)=2+43=63=2  (Answer)\begin{align*} \lim\limits_{x\to 2}\left(\frac{x^2+2x-8}{3x-6}\right)& =\lim\limits_{x\to 2}\left(\frac{\left(x+4\right)\left(x-2\right)}{3\left(x-2\right)}\right)\\ \\ &=\lim\limits_{x\to 2}\left(\frac{x+4}{3}\right)\\ \\ &=\frac{2+4}{3}\\ \\ &=\frac{6}{3}\\ \\ & =2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \\ \end{align*}
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