Question

the 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.

Explanation:

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

The graph has a filled point at $(2, -3)$, so $f(2) = -3$, meaning it is defined.

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

As $x$ approaches 2 from both left and right, the graph approaches the open point at $(2, 2)$. The left-hand limit and right-hand limit are equal, so the limit exists and $\lim_{x \to 2} f(x) = 2$.

Step3: Compare limit and $f(2)$

$\lim_{x \to 2} f(x) = 2$ and $f(2) = -3$, so $\lim_{x \to 2} f(x)
eq f(2)$.

Step4: Check continuity at $x=2$

A function is continuous at a point if $\lim_{x \to a} f(x) = f(a)$. Since this is not true here, the function is not continuous at $x=2$.

Answer:

• $f(2)$ is defined.
• $\lim_{x \to 2} f(x)$ exists.
• The function is not continuous at x = 2.