# Chapter 11 - Mid-Chapter Check Point - Page 1163: 5

The value of $\underset{x\to 0}{\mathop{\lim }}\,\left[ f\left( x \right)-g\left( x \right) \right]$ is $-1$.

#### Work Step by Step

To find the value of $\underset{x\to 0}{\mathop{\lim }}\,\left[ f\left( x \right)-g\left( x \right) \right]$, find the value of $\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)$ and the value of $\underset{x\to 0}{\mathop{\lim }}\,g\left( x \right)$ It is seen from the graph that as the value of x nears $0$ from the left or right, the value of the function $f\left( x \right)$ nears $0$. Thus, $\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)=0$ It is seen from the graph that as the value of x nears $0$ from the left or right, the value of the function $g\left( x \right)$ nears $1$. Thus, $\underset{x\to 0}{\mathop{\lim }}\,g\left( x \right)=1$ Now find the value of $\underset{x\to 0}{\mathop{\lim }}\,\left[ f\left( x \right)-g\left( x \right) \right]$, \begin{align} & \underset{x\to 0}{\mathop{\lim }}\,\left[ f\left( x \right)-g\left( x \right) \right]=\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)-\underset{x\to 0}{\mathop{\lim }}\,g\left( x \right) \\ & =0-1 \\ & =-1 \end{align} Thus, the value of $\underset{x\to 0}{\mathop{\lim }}\,\left[ f\left( x \right)-g\left( x \right) \right]$ is $-1$.

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