Answer
$\frac{1}{14}$
Work Step by Step
Numerator at $x=7:\,\,\,x-7=7-7=0$
Denominator at $x=7:\,\,\,x^{2}-49=7^{2}-49=0$
The function has the indeterminate form $\frac{0}{0}$ at $x=7$.
Transforming algebraically and canceling the common factor, we have
$\frac{x-7}{x^{2}-49}=\frac{x-7}{(x-7)(x+7)}=\frac{1}{x+7}$
Therefore,
$\lim\limits_{x \to 7}\frac{x-7}{x^{2}-49}=\lim\limits_{x \to 7}\frac{1}{x+7}=\frac{1}{7+7}=\frac{1}{14}$
