use the given graph of the function f to find the following limits: 1. $lim_{x\rightarrow1}f(x)=2$ help…

Explanation:

Step1: Recall limit definition

The limit of a function $f(x)$ as $x\rightarrow a$ exists if the left - hand limit $\lim_{x\rightarrow a^{-}}f(x)$ and the right - hand limit $\lim_{x\rightarrow a^{+}}f(x)$ are equal.

Step2: Analyze $\lim_{x\rightarrow1^{-}}f(x)$

As $x$ approaches $1$ from the left side (values of $x$ less than $1$), following the graph of the function, the $y$ - value approaches $2$. So, $\lim_{x\rightarrow1^{-}}f(x)=2$.

Step3: Analyze $\lim_{x\rightarrow1^{+}}f(x)$

As $x$ approaches $1$ from the right side (values of $x$ greater than $1$), following the graph of the function, the $y$ - value approaches $0$. So, $\lim_{x\rightarrow1^{+}}f(x)=0$.

Step4: Determine $\lim_{x\rightarrow1}f(x)$

Since $\lim_{x\rightarrow1^{-}}f(x)=2$ and $\lim_{x\rightarrow1^{+}}f(x)=0$, and $2\neq0$, $\lim_{x\rightarrow1}f(x)$ does not exist (DNE).

Step5: Analyze $\lim_{x\rightarrow4}f(x)$

As $x$ approaches $4$ from both the left and the right sides, the $y$ - value of the function approaches $0$. So, $\lim_{x\rightarrow4}f(x)=0$.

Answer:

1. DNE
2. 0
3. DNE
4. 0