Question

evaluate the limit, if it exists. (if an answer does not exist, enter
lim_{x
ightarrow6}\frac{\frac{1}{x}-\frac{1}{6}}{x - 6}

Explanation:

Step1: Combine the fractions in the numerator

First, find a common - denominator for $\frac{1}{x}-\frac{1}{6}$. The common denominator is $6x$. So, $\frac{1}{x}-\frac{1}{6}=\frac{6 - x}{6x}$.
The original limit becomes $\lim_{x
ightarrow6}\frac{\frac{6 - x}{6x}}{x - 6}$.

Step2: Simplify the complex - fraction

$\frac{\frac{6 - x}{6x}}{x - 6}=\frac{6 - x}{6x(x - 6)}$. Notice that $6 - x=-(x - 6)$. So, $\frac{6 - x}{6x(x - 6)}=\frac{-(x - 6)}{6x(x - 6)}$.

Step3: Cancel out the common factor

Cancel out the common factor $(x - 6)$ (for $x
eq6$). We get $\lim_{x
ightarrow6}\frac{-1}{6x}$.

Step4: Evaluate the limit

Substitute $x = 6$ into $\frac{-1}{6x}$. We have $\frac{-1}{6\times6}=-\frac{1}{36}$.

Answer:

$-\frac{1}{36}$