Question

question 15
let $f(x)=\

$$\begin{cases}8 - x - x^{2}&\\text{if }x\\leq1\\\\2x + 4&\\text{if }x>1\\end{cases}$$

$
calculate the following limits. enter \dne\ if the limit does not ex
$\lim\limits_{x\to1^{-}}f(x) = $
$\lim\limits_{x\to1^{+}}f(x) = $
$\lim\limits_{x\to1}f(x) = $
question help: video written example

Explanation:

Step1: Left limit, use $x\le1$ formula

Substitute $x=1$ into $8-x-x^2$:
$8 - 1 - 1^2 = 8-1-1=6$

Step2: Right limit, use $x>1$ formula

Substitute $x=1$ into $2x+4$:
$2(1) + 4 = 2+4=6$

Step3: Check two-sided limit validity

Since left limit = right limit = 6, the two-sided limit exists.

Answer:

$\lim_{x
ightarrow1^-} f(x) = 6$
$\lim_{x
ightarrow1^+} f(x) = 6$
$\lim_{x
ightarrow1} f(x) = 6$