record: 16 score: 16 for the function f, it is known that lim f(x) = lim f(x) and that x→3⁻ x→3⁺ lim f(x) =…

record: 16 score: 16 for the function f, it is known that lim f(x) = lim f(x) and that x→3⁻ x→3⁺ lim f(x) = lim f(x). which of the following must be x→3⁻ x→3⁺ true? i. f is continuous at x = 3. ii. f is differentiable at x = 3. i only both i and ii ii only neither i nor ii

record: 16 score: 16 for the function f, it is known that lim f(x) = lim f(x) and that x→3⁻ x→3⁺ lim f(x) = lim f(x). which of the following must be x→3⁻ x→3⁺ true? i. f is continuous at x = 3. ii. f is differentiable at x = 3. i only both i and ii ii only neither i nor ii

Answer

Explanation:

Step1: Recall continuity condition

A function $f(x)$ is continuous at $x = a$ if $\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a^{+}}f(x)=f(a)$. Given $\lim_{x\rightarrow 3^{-}}f(x)=\lim_{x\rightarrow 3^{+}}f(x)$, but we don't know if it equals $f(3)$. So we can't be sure of continuity.

Step2: Recall differentiability condition

A function $f(x)$ is differentiable at $x = a$ if $\lim_{x\rightarrow a^{-}}f^{\prime}(x)=\lim_{x\rightarrow a^{+}}f^{\prime}(x)$ and the function is continuous at $x = a$. We only know $\lim_{x\rightarrow 3^{-}}f^{\prime}(x)=\lim_{x\rightarrow 3^{+}}f^{\prime}(x)$, and we are not sure about continuity at $x = 3$. So we can't be sure of differentiability.

Answer:

Neither I nor II