Question

exercises 2.2 the limit of a function
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let $f(x)=\begin{cases}10 - x - x^{2}&\text{if }xleq3\\2x - 8&\text{if }x > 3end{cases}$
calculate the following limits. enter \dne\ if the limit does not exist.
$lim_{x
ightarrow3^{-}}f(x)=-12$
$lim_{x
ightarrow3^{+}}f(x)=4$
$lim_{x
ightarrow3}f(x)=dne$
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Explanation:

Step1: Calculate left - hand limit

For $x\to3^-$, use $f(x)=10 - x - x^{2}$. Substitute $x = 3$ into $10 - x - x^{2}$:
$10-3 - 3^{2}=10 - 3-9=-2$.

Step2: Calculate right - hand limit

For $x\to3^+$, use $f(x)=2x - 8$. Substitute $x = 3$ into $2x - 8$:
$2\times3 - 8=6 - 8=-2$.

Step3: Determine the overall limit

Since $\lim_{x\to3^-}f(x)=\lim_{x\to3^+}f(x)=-2$, then $\lim_{x\to3}f(x)=-2$.

Answer:

$\lim_{x\to3^-}f(x)=-2$
$\lim_{x\to3^+}f(x)=-2$
$\lim_{x\to3}f(x)=-2$