Questions: 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.

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.

Transcript text: 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. $\lim _{x \rightarrow 2} f(x)$ exists. $\lim _{x \rightarrow 2} f(x)=f(2)$ The function is continuous at $\mathrm{x}=2$. The function is not continuous at $\mathrm{x}=2$.

Solution

Select all statements below that you agree with. Determine $f(2)$ $f(2) = 4$ since the solid dot at $x = 2$ is at $y=4$. So, $f(2)$ is defined. Determine if $\lim_{x \rightarrow 2} f(x)$ exists. As $x$ approaches 2 from the left, $f(x)$ approaches 3. As $x$ approaches 2 from the right, $f(x)$ approaches 3. Thus, $\lim_{x \rightarrow 2} f(x) = 3$. So $\lim_{x \rightarrow 2} f(x)$ exists. Determine if $\lim_{x \rightarrow 2} f(x) = f(2)$ $\lim_{x \rightarrow 2} f(x) = 3$ and $f(2) = 4$. Since $3 \ne 4$, $\lim_{x \rightarrow 2} f(x) \ne f(2)$. Determine if the function is continuous at $x = 2$. A function is continuous at $x = c$ if $f(c)$ is defined, $\lim_{x \rightarrow c} f(x)$ exists, and $\lim_{x \rightarrow c} f(x) = f(c)$. Since $\lim_{x \rightarrow 2} f(x) \ne f(2)$, the function is not continuous at $x = 2$.

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

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