Question

score on last try: 0 of 1 pts. see details for more. > next question get a similar question you can retry this question below 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

Looking at the graph, there is an open - circle at \(x = 2\) on the curve of \(y=f(x)\) and a filled - circle at \((2,-4)\). The value of the function at \(x = 2\) is given by the filled - circle. So \(f(2)=-4\), and \(f(2)\) is defined.

Step2: Check if \(\lim_{x

ightarrow2}f(x)\) exists
As \(x\) approaches \(2\) from the left and from the right, the function values approach the same \(y\) - value (the value that the curve seems to be approaching near \(x = 2\) which is \(3\)). So \(\lim_{x
ightarrow2}f(x)\) exists.

Step3: Check if \(\lim_{x

ightarrow2}f(x)=f(2)\)
We found that \(\lim_{x
ightarrow2}f(x) = 3\) and \(f(2)=-4\). So \(\lim_{x
ightarrow2}f(x)
eq f(2)\)

Step4: Check continuity at \(x = 2\)

A function \(y = f(x)\) is continuous at \(x=a\) if \(\lim_{x
ightarrow a}f(x)=f(a)\). 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 \(x = 2\)