find the limit.\n$$\\lim_{y \\to 8} \\frac{\\frac{1}{y} - \\frac{1}{8}}{y - 8}$$\nselect the correct choice…

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

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

Answer

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\rightarrow8}\frac{\frac{1}{y}-\frac{1}{8}}{y - 8}=\lim_{y\rightarrow8}\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\neq8)).

Step3: Evaluate the limit

Now, find (\lim_{y\rightarrow8}-\frac{1}{8y}). Substitute (y = 8) into (-\frac{1}{8y}). We get (-\frac{1}{8\times8}=-\frac{1}{64}).

Answer:

A. (\lim_{y\rightarrow8}\frac{\frac{1}{y}-\frac{1}{8}}{y - 8}=-\frac{1}{64})