Question

find the limit.
$$\lim_{y \to 8} \frac{\frac{1}{y} - \frac{1}{8}}{y - 8}$$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $$\lim_{y \to 8} \frac{\frac{1}{y} - \frac{1}{8}}{y - 8} = $$ (type an integer or a simplified fraction.)
b. the limit does not exist.

Explanation:

Step1: Simplify the numerator

First, find a common denominator for \(\frac{1}{y}-\frac{1}{8}\). The common denominator of \(y\) and \(8\) is \(8y\). So \(\frac{1}{y}-\frac{1}{8}=\frac{8 - y}{8y}\).
Then the original limit \(\lim_{y
ightarrow8}\frac{\frac{1}{y}-\frac{1}{8}}{y - 8}=\lim_{y
ightarrow8}\frac{\frac{8 - y}{8y}}{y - 8}\).

Step2: Simplify the fraction

Since \(y-8=-(8 - y)\), the fraction \(\frac{\frac{8 - y}{8y}}{y - 8}=\frac{8 - y}{8y(y - 8)}=-\frac{1}{8y}\) (for \(y
eq8\)).

Step3: Evaluate the limit

Now, find \(\lim_{y
ightarrow8}-\frac{1}{8y}\). Substitute \(y = 8\) into \(-\frac{1}{8y}\). We get \(-\frac{1}{8\times8}=-\frac{1}{64}\).

Answer:

A. \(\lim_{y
ightarrow8}\frac{\frac{1}{y}-\frac{1}{8}}{y - 8}=-\frac{1}{64}\)