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.

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.

Answer

Explanation:

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

The graph has a filled - in dot at (x = 2) and (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 values approach (2), and as (x) approaches (2) from the right, the function values approach (2). So (\lim_{x\rightarrow2}f(x) = 2) exists.

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

We have (\lim_{x\rightarrow2}f(x) = 2) and (f(2)=3). So (\lim_{x\rightarrow2}f(x)\neq f(2)).

Step4: Check continuity

A function (y = f(x)) is continuous at (x=a) if (\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).