Evaluate: \lim_{x arrow (1)/(5)^+} ((x)/(1-5x))

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Expression: $\lim_{x \rightarrow \frac{ 1 }{ 5 }^+} \left(\frac{ x }{ 1-5x }\right)$

Write the expression as a product with the factor $x$

$\lim_{x \rightarrow \frac{ 1 }{ 5 }^+} \left(x \times \frac{ 1 }{ 1-5x }\right)$

Evaluate the limit of each term separately

$\begin{array} { l }\lim_{x \rightarrow \frac{ 1 }{ 5 }^+} \left(x\right),\\\lim_{x \rightarrow \frac{ 1 }{ 5 }^+} \left(\frac{ 1 }{ 1-5x }\right)\end{array}$

Evaluate the limit

$\begin{array} { l }\frac{ 1 }{ 5 },\\\lim_{x \rightarrow \frac{ 1 }{ 5 }^+} \left(\frac{ 1 }{ 1-5x }\right)\end{array}$

Evaluate the limit

$\begin{array} { l }\frac{ 1 }{ 5 },\\-\infty\end{array}$

Since the expression $\begin{array} { l }a \times \left( -\infty \right),& a > 0\end{array}$ is defined as $-\infty$, the limit $\lim_{x \rightarrow \frac{ 1 }{ 5 }^+} \left(x \times \frac{ 1 }{ 1-5x }\right)$ equals $-\infty$

$-\infty$

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