Question

score on last try: 0 of 1 pt. see details for more.get a similar question you can retry this question belowthe graph below is the function $f(x)$select all statements below that you agree with.note: you may be checking more than one box. no partial credit.$square$ $f(2)$ is defined.$square$ $lim_{x \to 2} f(x)$ exists.$square$ $lim_{x \to 2} f(x) = f(2)$.$square$ the function is continuous at x = 2.$square$ the function is not continuous at x = 2.question help: video

Explanation:

Step1: Check if $f(2)$ is defined

At $x=2$, there is a filled black dot at $y=-1$, so $f(2)=-1$, meaning it is defined.

Step2: Check if $\lim_{x \to 2} f(x)$ exists

As $x$ approaches 2 from the left, the graph approaches $y=-1$. As $x$ approaches 2 from the right, the graph approaches $y=-2$. Since the left-hand limit $\lim_{x \to 2^-} f(x) = -1$ and the right-hand limit $\lim_{x \to 2^+} f(x) = -2$ are not equal, the two-sided limit does not exist.

Step3: Check $\lim_{x \to 2} f(x) = f(2)$

Since $\lim_{x \to 2} f(x)$ does not exist, this equality cannot hold.

Step4: Check continuity at $x=2$

A function is continuous at a point if: 1) $f(a)$ is defined, 2) $\lim_{x \to a} f(x)$ exists, 3) $\lim_{x \to a} f(x) = f(a)$. Since the limit does not exist, the function is not continuous at $x=2$.

Answer:

• $f(2)$ is defined.
• The function is not continuous at x = 2.