Question

For \[f(x)=|x|=\begin{cases}-x,&x<0\\x,&x\geq0\end{cases}\], what are the LHL, RHL, and \[f(0)\] at \[x=0\]?

Options

• \[\text{LHL}=\text{RHL}=f(0)=0\]

• \[\text{LHL}=\text{RHL}=0,\ f(0)=1\]

• \[\text{LHL}=-1,\ \text{RHL}=1,\ f(0)=0\]

• \[\text{LHL}=0,\ \text{RHL}=1,\ f(0)=0\]

Answer 1

As \[x\] approaches \[0\] from either side, \[|x|\] approaches \[0\]. Since both one-sided limits equal \[f(0)=0\], \[f(x)=|x|\] is continuous at \[x=0\].