Differential and Integral Calculus by Feliciano and Uy, Exercise 1.3, Problem 10

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PROBLEM:

Evaluate lim⁡x→2(x3−8x2−4 )\displaystyle \lim\limits_{x\to 2}\left(\frac{x^3-8}{x^2-4\:}\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(x3−8x2−4 )=lim⁡x→2[(x−2)(x2+2x+4)(x+2)(x−2)]=lim⁡x→2[(x2+2x+4)(x+2)]=22+2⋅2+42+2=3  (Answer)\begin{align*} \lim\limits_{x\to 2}\left(\frac{x^3-8}{x^2-4\:}\right) & =\lim\limits_{x\to 2}\left[\frac{\left(x-2\right)\left(x^2+2x+4\right)}{\left(x+2\right)\left(x-2\right)}\right] \\ &=\lim\limits_{x\to 2}\left[\frac{\left(x^2+2x+4\right)}{\left(x+2\right)}\right] \\ & =\frac{2^2+2\cdot 2+4}{2+2} \\ & =3 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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