Question

the graph of the function f is shown in the figure above. which of the following statements must be false? a $lim_{x
ightarrow2}f(x)=3$ b $lim_{x
ightarrow2}f(x)=infty$ c $lim_{x
ightarrow2}f(x)=f(2)$ d $lim_{x
ightarrowinfty}f(x)=0$

Explanation:

Step1: Recall limit - definition

The limit $\lim_{x
ightarrow a}f(x)$ exists if and only if the left - hand limit $\lim_{x
ightarrow a^{-}}f(x)$ and the right - hand limit $\lim_{x
ightarrow a^{+}}f(x)$ exist and are equal.

Step2: Analyze the graph near $x = 2$

As $x$ approaches $2$ from the left, $f(x)$ approaches $+\infty$, and as $x$ approaches $2$ from the right, $f(x)$ approaches $0$. So, $\lim_{x
ightarrow 2}f(x)$ does not exist. Also, $f(2)$ is a finite value (the $y$ - value of the function at $x = 2$). Since the limit $\lim_{x
ightarrow 2}f(x)$ does not exist, the statement $\lim_{x
ightarrow 2}f(x)=f(2)$ must be false.

Step3: Analyze other options

• Option A: $\lim_{x

ightarrow 2}f(x)=3$ is false because the one - sided limits at $x = 2$ are not equal to $3$.

• Option B: $\lim_{x

ightarrow 2}f(x)=\infty$ is not correct in the strict sense of a two - sided limit, but the left - hand limit as $x
ightarrow 2^{-}$ is $\infty$.

• Option D: $\lim_{x

ightarrow\infty}f(x)=0$ seems reasonable as the function appears to approach $y = 0$ as $x$ gets large.

Answer:

C. $\lim_{x
ightarrow 2}f(x)=f(2)$