Question

select the correct answer below. o f(x) has a removable discontinuity at x = 3. o f(x) has a jump discontinuity at x = 3. o f(x) has an infinite discontinuity at x = 3. o f(x) is continuous at x = 3.

Explanation:

Step1: Check continuity definition

A function is continuous at a point if $\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)=f(a)$. A removable discontinuity occurs when the left - hand limit and the right - hand limit exist and are equal, but the function is not defined or has a different value at that point. A jump discontinuity occurs when $\lim_{x
ightarrow a^{-}}f(x)
eq\lim_{x
ightarrow a^{+}}f(x)$. An infinite discontinuity occurs when either $\lim_{x
ightarrow a^{-}}f(x)=\pm\infty$ or $\lim_{x
ightarrow a^{+}}f(x)=\pm\infty$.

Step2: Analyze the graph at $x = 3$

Looking at the graph of $y = f(x)$ at $x = 3$, we see that the left - hand limit and the right - hand limit as $x$ approaches 3 are not equal. The function approaches different values from the left and from the right of $x = 3$.

Answer:

f(x) has a jump discontinuity at x = 3.