Question

use the graph of f to complete parts (a) through (l). if a limit does not exist, explain why. (k) find lim f(x). select the correct choice below, and fill in the answer box if necessary. x→2+ a. lim f(x)= 3 x→2+ (type an integer or a fraction.) b. the limit does not exist because f(x) is not defined for all x > 2. (l) find lim f(x). select the correct choice below, and fill in the answer box if necessary. x→2 a. lim f(x)= x→2 (type an integer or a fraction.) b. the limit does not exist because lim f(x)≠lim f(x) x→2− x→2+

Explanation:

Step1: Recall the definition of the limit

The limit $\lim_{x
ightarrow a}f(x)$ exists if and only if $\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)$. We already know from part (k) that $\lim_{x
ightarrow 2^{+}}f(x) = 3$. We need to find $\lim_{x
ightarrow 2^{-}}f(x)$ by looking at the graph as $x$ approaches 2 from the left - hand side.

Step2: Analyze the left - hand limit

As $x$ approaches 2 from the left - hand side ($x
ightarrow 2^{-}$), the $y$ - values of the function $f(x)$ approach 2.

Step3: Compare left - hand and right - hand limits

Since $\lim_{x
ightarrow 2^{-}}f(x)=2$ and $\lim_{x
ightarrow 2^{+}}f(x)=3$, we have $\lim_{x
ightarrow 2^{-}}f(x)
eq\lim_{x
ightarrow 2^{+}}f(x)$.

Answer:

B. The limit does not exist because $\lim_{x
ightarrow 2^{-}}f(x)
eq\lim_{x
ightarrow 2^{+}}f(x)$