Question

for the function $f(x)$ shown below, determine $lim_{x
ightarrow4}f(x)$. $f(x)=\begin{cases}-x - 2&\text{for }x>4\\-x^{2}+10&\text{for }xleq4end{cases}$ answer attempt 1 out of 2 dne

Explanation:

Step1: Find left - hand limit

We use the part of the function for $x\leq4$. So, $\lim_{x
ightarrow4^{-}}f(x)=\lim_{x
ightarrow4^{-}}(-x^{2}+10)$. Substitute $x = 4$ into $-x^{2}+10$: $-(4)^{2}+10=-16 + 10=-6$.

Step2: Find right - hand limit

We use the part of the function for $x>4$. So, $\lim_{x
ightarrow4^{+}}f(x)=\lim_{x
ightarrow4^{+}}(-x - 2)$. Substitute $x = 4$ into $-x - 2$: $-4-2=-6$.

Step3: Determine the limit

Since $\lim_{x
ightarrow4^{-}}f(x)=\lim_{x
ightarrow4^{+}}f(x)=-6$, then $\lim_{x
ightarrow4}f(x)=-6$.

Answer:

$-6$