Question

let (f(x)=\begin{cases}2 - x - x^{2}&\text{if }xleq2\\2x - 5&\text{if }x>2end{cases}
calculate the following limits. enter \dne\ if the limit does not exist.
(lim_{x
ightarrow2^{-}}f(x)=)
(lim_{x
ightarrow2^{+}}f(x)=)
(lim_{x
ightarrow2}f(x)=)
question help: video message instructor

Explanation:

Step1: Find left - hand limit

For $\lim_{x
ightarrow2^{-}}f(x)$, since $x
ightarrow2^{-}$ means $x < 2$, we use the function $f(x)=2 - x - x^{2}$. Substitute $x = 2$ into $2 - x - x^{2}$:

$$
LATEXBLOCK0
$$

Step2: Find right - hand limit

For $\lim_{x
ightarrow2^{+}}f(x)$, since $x
ightarrow2^{+}$ means $x>2$, we use the function $f(x)=2x - 5$. Substitute $x = 2$ into $2x - 5$:

$$
LATEXBLOCK1
$$

Step3: Determine the overall limit

Since $\lim_{x
ightarrow2^{-}}f(x)=-4$ and $\lim_{x
ightarrow2^{+}}f(x)=-1$, and $\lim_{x
ightarrow2^{-}}f(x)
eq\lim_{x
ightarrow2^{+}}f(x)$, then $\lim_{x
ightarrow2}f(x)$ does not exist (DNE).

Answer:

$\lim_{x
ightarrow2^{-}}f(x)=-4$
$\lim_{x
ightarrow2^{+}}f(x)=-1$
$\lim_{x
ightarrow2}f(x)=\text{DNE}$