Questions: The function f is graphed below. Find the following limits. If a limit does not exist, click on "Does Not Exist." (a) lim x → 4− f(x)= □ (b) lim x → 4+ f(x)= □ (c) lim x → 4 f(x)= □

Transcript text: The function $f$ is graphed below. Find the following limits. If a limit does not exist, click on "Does Not Exist." (a) $\lim _{x \rightarrow 4^{-}} f(x)=$ $\square$ (b) $\lim _{x \rightarrow 4^{+}} f(x)=$ $\square$ (c) $\lim _{x \rightarrow 4} f(x)=$ $\square$

Solution

Solution Steps

Step 1: Evaluate the left-hand limit as x approaches 4

To find $\lim_{{x \to 4^-}} f(x)$, observe the value that $f(x)$ approaches as $x$ approaches 4 from the left. From the graph, as $x$ approaches 4 from the left, $f(x)$ approaches 2.

Step 2: Evaluate the right-hand limit as x approaches 4

To find $\lim_{{x \to 4^+}} f(x)$, observe the value that $f(x)$ approaches as $x$ approaches 4 from the right. From the graph, as $x$ approaches 4 from the right, $f(x)$ approaches -2.

Step 3: Determine the overall limit as x approaches 4

To find $\lim_{{x \to 4}} f(x)$, compare the left-hand limit and the right-hand limit. Since $\lim_{{x \to 4^-}} f(x) = 2$ and $\lim_{{x \to 4^+}} f(x) = -2$, the left-hand limit and the right-hand limit are not equal. Therefore, the overall limit does not exist.