# How do you find the limit of  ((1/y) - (1/7))/(y-7) as y approaches 7?

Feb 4, 2016

$- \frac{1}{49}$

#### Explanation:

This can be expressed as

${\lim}_{y \rightarrow 7} \frac{\frac{1}{y} - \frac{1}{7}}{y - 7}$

We should try to clear the fractions from the denominator by multiplying the fraction by $\frac{7 y}{7 y}$.

$= {\lim}_{y \rightarrow 7} \frac{\frac{7 y}{y} - \frac{7 y}{7}}{7 y \left(y - 7\right)}$

This simplifies to be

$= {\lim}_{y \rightarrow 7} \frac{7 - y}{7 y \left(y - 7\right)}$

Note that this can be simplified, since $7 - y = - \left(y - 7\right)$.

$= {\lim}_{y \rightarrow 7} \frac{- \left(y - 7\right)}{7 y \left(y - 7\right)}$

The $y - 7$ terms will cancel.

$= {\lim}_{y \rightarrow 7} \frac{- 1}{7 y}$

The limit can now be evaluated by plugging in $7$ for $y$.

$= \frac{- 1}{7 \left(7\right)}$

$= - \frac{1}{49}$

We can check the graph of the function. Even though the point at $7$ is undefined, it should be at $- \frac{1}{49}$ (very close to 0).

graph{(1/x-1/7)/(x-7) [6.347, 7.6793, -0.3486, 0.317]}