Differential and Integral Calculus by Feliciano and Uy, Exercise 1.3, Problem 9

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PROBLEM:

Evaluate lim⁡x→4(1x−14x−4 )\displaystyle \lim\limits_{x\to 4}\left(\frac{\frac{1}{x}-\frac{1}{4}}{x-4\:}\right).

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SOLUTION:

A straight substitution of x=4x=4 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→4(1x−14x−4)=lim⁡x→4(4−x4xx−4)=lim⁡x→44−x4x(x−4)=lim⁡x→4(4−x−4x(4−x))=lim⁡x→4−14x=−14⋅4=−116  (Answer)\begin{align*} \\ \lim\limits_{x\to 4}\left(\frac{\frac{1}{x}-\frac{1}{4}}{x-4}\right)& =\lim\limits_{x\to 4}\left(\frac{\frac{4-x}{4x}}{x-4}\right)\\ \\ & =\lim\limits_{x\to 4}\frac{4-x}{4x\left(x-4\right)}\\ \\ &=\lim\limits_{x\to 4}\left(\frac{4-x}{-4x\left(4-x\right)}\right)\\ \\ & =\lim\limits_{x\to 4}-\frac{1}{4x}\\ \\ & =-\frac{1}{4\cdot 4}\\ \\ & =-\frac{1}{16} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \\ \end{align*}
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