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Limit of a Function in Indeterminate Form| Differential and Integral Calculus| Feliciano and Uy| Exercise 1.3, Problem 9|

Evaluate

\lim\limits_{x\to 4}\left(\frac{\frac{1}{x}-\frac{1}{4}}{x-4\:}\right)

 

SOLUTION

A straight substitution of  x=4 leads to the indeterminate form   \frac{0}{0}   which is meaningless.

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

\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}