determine for what numbers, if any, the given function is discontinuous. f(x) = { x + 8 if x ≤ 0; 8 if 0 < x…
determine for what numbers, if any, the given function is discontinuous. f(x) = { x + 8 if x ≤ 0; 8 if 0 < x ≤ 8; x² + 8 if x > 8. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the function is discontinuous at x = (use a comma to separate answers as needed.) b. the function is continuous everywhere.
Answer
Explanation:
Step1: Check continuity at $x = 0$
Left - hand limit: $\lim_{x\rightarrow0^{-}}f(x)=\lim_{x\rightarrow0^{-}}(x + 8)=0 + 8=8$. Right - hand limit: $\lim_{x\rightarrow0^{+}}f(x)=\lim_{x\rightarrow0^{+}}8 = 8$. $f(0)=0 + 8=8$. Since $\lim_{x\rightarrow0^{-}}f(x)=\lim_{x\rightarrow0^{+}}f(x)=f(0)$, the function is continuous at $x = 0$.
Step2: Check continuity at $x = 8$
Left - hand limit: $\lim_{x\rightarrow8^{-}}f(x)=\lim_{x\rightarrow8^{-}}8 = 8$. Right - hand limit: $\lim_{x\rightarrow8^{+}}f(x)=\lim_{x\rightarrow8^{+}}(x^{2}+8)=8^{2}+8=64 + 8=72$. Since $\lim_{x\rightarrow8^{-}}f(x)\neq\lim_{x\rightarrow8^{+}}f(x)$, the function is discontinuous at $x = 8$.
Answer:
A. The function is discontinuous at $x = 8$