the graph below is the function f(x)\nselect all statements below that you agree with. note: you may be…

the graph below is the function f(x)\nselect all statements below that you agree with. note: you may be checking more than one box. no partial credit.\n□f(2) is defined.\n□lim_{x→2} f(x) exists.\n□lim_{x→2} f(x)=f(2).\n□the function is continuous at x = 2.\n□the function is not continuous at x = 2.\nquestion help: □video □message instructor

the graph below is the function f(x)\nselect all statements below that you agree with. note: you may be checking more than one box. no partial credit.\n□f(2) is defined.\n□lim_{x→2} f(x) exists.\n□lim_{x→2} f(x)=f(2).\n□the function is continuous at x = 2.\n□the function is not continuous at x = 2.\nquestion help: □video □message instructor

Answer

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 $\lim_{x\rightarrow2}f(x)$ 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 $\lim_{x\rightarrow2}f(x)$ exists.

Step3: Check if $\lim_{x\rightarrow2}f(x)=f(2)$

$\lim_{x\rightarrow2}f(x)$ 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 $x = a$, $\lim_{x\rightarrow a}f(x)=f(a)$. Since $\lim_{x\rightarrow2}f(x)\neq f(2)$, the function is not continuous at $x = 2$.

Answer:

$f(2)$ is defined. $\lim_{x\rightarrow2}f(x)$ exists. The function is not continuous at $x = 2$.