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.
□f(2) is defined.
□lim_{x→2} f(x) exists.
□lim_{x→2} f(x)=f(2).
□the function is continuous at x = 2.
□the function is not continuous at x = 2.
question help: □video □message instructor

Explanation:

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

The graph has a filled - in dot at $x = 2$ with $y=3$, so $f(2)=3$ and $f(2)$ is defined.

Step2: Check if $\lim_{x

ightarrow2}f(x)$ exists
As $x$ approaches 2 from the left, the function approaches the open - circle value. As $x$ approaches 2 from the right, the function approaches the same open - circle value. So $\lim_{x
ightarrow2}f(x)$ exists.

Step3: Check if $\lim_{x

ightarrow2}f(x)=f(2)$
$\lim_{x
ightarrow2}f(x)$ is the value of the open - circle and $f(2)$ is the value of the filled - in dot. They are not equal.

Step4: Check continuity

For a function to be continuous at $x = a$, $\lim_{x
ightarrow a}f(x)=f(a)$. Since $\lim_{x
ightarrow2}f(x)
eq f(2)$, the function is not continuous at $x = 2$.

Answer:

$f(2)$ is defined.
$\lim_{x
ightarrow2}f(x)$ exists.
The function is not continuous at $x = 2$.