Problem
Evaluate the following limits. \[ f(x)=\left\{\begin{array}{ll} -x+6, & x<0 \\ x+6, & x \geq 0 \end{array}\right. \] Show your work here \[ \lim _{x \rightarrow 0^{-}} f(x)= \] \[ \lim _{x \rightarrow 0^{+}} f(x)= \] \[ \lim _{x \rightarrow 0} f(x)= \]
Solution
Step 1 ：Evaluate the limit from the left: \(\lim_{x \rightarrow 0^{-}} f(x) = -0 + 6 = 6\)
Step 2 ：Evaluate the limit from the right: \(\lim_{x \rightarrow 0^{+}} f(x) = 0 + 6 = 6\)
Step 3 ：Since the limits from the left and right are equal, the overall limit exists: \(\lim_{x \rightarrow 0} f(x) = \boxed{6}\)
