Consider
\[ g(x) = \begin{cases} -2e^x & \text {if }x<0\\ \\ \frac {x-6}{x+3} & \text {if }x\geq 0 \end{cases} \]
We will use the definition of continuity to determine if \(g\) is continuous at 0. Calculate the following (Use DNE if a limit Does Not Exist.):
\begin{align*} \lim _{x\to 0^-}g(x) &= \answer {-2}\\ \lim _{x\to 0^+}g(x) &= \answer {-2} \end{align*}
Thus
\[ \lim _{x\to 0}g(x) = \answer {-2} \]
This verifies that \(\displaystyle \lim _{x\to 0} g(x)\) exists.
Calculate the value of \(g(0)= \answer {-2}\).
This verified that the value of \(g(0)\) exists.
Based on the above work we conclude that \(g\)
is continuous is not continuous
at
\(x=0\).
The interval of continuity for \(g\) is \(\left (\answer {-\infty }\, , \,\answer {\infty }\right )\).