does $lim_{x\rightarrow8}f(x)$ exist? yes no
does $lim_{x\rightarrow8}f(x)$ exist? yes no
Answer
Explanation:
Step1: Recall limit - existence condition
The limit $\lim_{x\rightarrow a}f(x)$ exists if and only if $\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a^{+}}f(x)$. Here $a = 8$.
Step2: Analyze left - hand limit
As $x$ approaches $8$ from the left ($x\rightarrow8^{-}$), we follow the graph of the function for values of $x$ less than $8$. The function approaches $- 8$.
Step3: Analyze right - hand limit
As $x$ approaches $8$ from the right ($x\rightarrow8^{+}$), we follow the graph of the function for values of $x$ greater than $8$. The function approaches $-10$.
Step4: Compare left - hand and right - hand limits
Since $\lim_{x\rightarrow8^{-}}f(x)\neq\lim_{x\rightarrow8^{+}}f(x)$ (because $-8\neq - 10$), the limit $\lim_{x\rightarrow8}f(x)$ does not exist.
Answer:
no