question for the function f(x) shown below, determine lim f(x). x→ - 4 f(x) = {6 + 2x for x < - 4 {10 - x²…

Explanation:

Step1: Find left - hand limit

We find $\lim_{x\rightarrow - 1^{-}}f(x)$. Since $x\rightarrow - 1^{-}$ means $x < - 1$, we use $f(x)=6 + 2x$. $\lim_{x\rightarrow - 1^{-}}f(x)=\lim_{x\rightarrow - 1}(6 + 2x)=6+2\times(-1)=4$

Step2: Find right - hand limit

We find $\lim_{x\rightarrow - 1^{+}}f(x)$. Since $x\rightarrow - 1^{+}$ means $x > - 1$, we use $f(x)=10 - x^{2}$. $\lim_{x\rightarrow - 1^{+}}f(x)=\lim_{x\rightarrow - 1}(10 - x^{2})=10-(-1)^{2}=9$

Step3: Check if limit exists

Since $\lim_{x\rightarrow - 1^{-}}f(x)=4$ and $\lim_{x\rightarrow - 1^{+}}f(x)=9$, and $\lim_{x\rightarrow - 1^{-}}f(x)\neq\lim_{x\rightarrow - 1^{+}}f(x)$, the limit $\lim_{x\rightarrow - 1}f(x)$ does not exist.

Answer:

The limit $\lim_{x\rightarrow - 1}f(x)$ does not exist.