Evaluate the following limits.
\[
f(x)=\left\{\begin{array}{ll}
-x+6, & x< 0 \\
x+6, & x \geq 0
\end{array}\right.
\]
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\[
\lim _{x \rightarrow 0^{-}} f(x)=
\]
\[
\lim _{x \rightarrow 0^{+}} f(x)=
\]
\[
\lim _{x \rightarrow 0} f(x)=
\]

Answer

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Answer

Since the limits from the left and right are equal, the overall limit exists: \(\lim_{x \rightarrow 0} f(x) = \boxed{6}\)

Steps

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: \(\lim_{x \rightarrow 0} f(x) = \boxed{6}\)