Answer
The value of $\underset{x\to 0}{\mathop{\lim }}\,\frac{\frac{1}{x+10}-\frac{1}{10}}{x}$ is $\frac{-1}{100}$.
Work Step by Step
Consider the function $f\left( x \right)=\frac{\frac{1}{x+10}-\frac{1}{10}}{x}$ ,
Simplify the above function,
$\begin{align}
& f\left( x \right)=\frac{\frac{1}{x+10}-\frac{1}{10}}{x} \\
& =\frac{\frac{10-\left( x+10 \right)}{10\left( x+10 \right)}}{x} \\
& =\frac{\frac{10-x-10}{10\left( x+10 \right)}}{x} \\
& =\frac{-x}{10x\left( x+10 \right)}
\end{align}$
$=\frac{-1}{10\left( x+10 \right)}$
Since, the function $g\left( x \right)=x+10$ is a polynomial,
Find the value of $\underset{x\to 0}{\mathop{\lim }}\,\frac{\frac{1}{x+10}-\frac{1}{10}}{x}$ ,
$\begin{align}
& \underset{x\to 0}{\mathop{\lim }}\,\frac{\frac{1}{x+10}-\frac{1}{10}}{x}=\underset{x\to 0}{\mathop{\lim }}\,\frac{-1}{10\left( x+10 \right)} \\
& =\frac{-1}{10\left( \underset{x\to 0}{\mathop{\lim }}\,x+10 \right)} \\
& =\frac{-1}{10\left( 0+10 \right)} \\
& =\frac{-1}{10\left( 10 \right)}
\end{align}$
$=\frac{-1}{100}$
Thus, the value of $\underset{x\to 0}{\mathop{\lim }}\,\frac{\frac{1}{x+10}-\frac{1}{10}}{x}$ is $\frac{-1}{100}$.
