Question

the graph of f is shown below. answer the following:
a. \\(\lim_{x\to3}f(x)=\\)
b. \\(\lim_{x\to5^{-}}f(x)=\\)
c. \\(\lim_{x\to5}f(x)=\\)
f(5)=undefined

Explanation:

Step1: Recall limit definition

The limit $\lim_{x
ightarrow a}f(x)$ is the value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$. For $\lim_{x
ightarrow 3}f(x)$, as $x$ approaches 3 from both the left - hand side and the right - hand side of $x = 3$, the function values approach 0.

Step2: Analyze left - hand limit at $x = 5$

The left - hand limit $\lim_{x
ightarrow 5^{-}}f(x)$ is the value that $f(x)$ approaches as $x$ approaches 5 from the left. Looking at the graph, as $x$ approaches 5 from the left, $f(x)$ approaches 1.

Step3: Analyze right - hand limit at $x = 5$

The right - hand limit $\lim_{x
ightarrow 5^{+}}f(x)$ is the value that $f(x)$ approaches as $x$ approaches 5 from the right. From the graph, as $x$ approaches 5 from the right, $f(x)$ approaches 4. Since the left - hand limit and the right - hand limit at $x = 5$ are not equal, $f(5)$ is undefined.

Answer:

a. $\lim_{x
ightarrow 3}f(x)=0$
b. $\lim_{x
ightarrow 5^{-}}f(x)=1$
c. $\lim_{x
ightarrow 5^{+}}f(x)=4$
d. $f(5)$ is undefined