Evaluate: \lim_{y arrow 5} ((y-\frac{25)/(y)}{y-5})

Expression: $\lim_{y \rightarrow 5} \left(\frac{ y-\frac{ 25 }{ y } }{ y-5 }\right)$

Evaluate the limits of numerator and denominator separately

$\begin{array} { l }\lim_{y \rightarrow 5} \left(y-\frac{ 25 }{ y }\right),\\\lim_{y \rightarrow 5} \left(y-5\right)\end{array}$

Evaluate the limit

$\begin{array} { l }0,\\\lim_{y \rightarrow 5} \left(y-5\right)\end{array}$

Evaluate the limit

$\begin{array} { l }0,\\0\end{array}$

Since the expression $\frac{ 0 }{ 0 }$ is an indeterminate form, try transforming the expression

$\lim_{y \rightarrow 5} \left(\frac{ y-\frac{ 25 }{ y } }{ y-5 }\right)$

Write all numerators above the common denominator

$\lim_{y \rightarrow 5} \left(\frac{ \frac{ {y}^{2}-25 }{ y } }{ y-5 }\right)$

Simplify the complex fraction

$\lim_{y \rightarrow 5} \left(\frac{ {y}^{2}-25 }{ y \times \left( y-5 \right) }\right)$

Use ${a}^{2}-{b}^{2}=\left( a-b \right)\left( a+b \right)$ to factor the expression

$\lim_{y \rightarrow 5} \left(\frac{ \left( y-5 \right) \times \left( y+5 \right) }{ y \times \left( y-5 \right) }\right)$

Cancel out the common factor $y-5$

$\lim_{y \rightarrow 5} \left(\frac{ y+5 }{ y }\right)$

Evaluate the limit

$\frac{ 5+5 }{ 5 }$

Simplify the expression

$2$

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