use the given graph of the function f to find the following limits:\n1. $lim_{x\rightarrow1}f(x)=$ dne help…

Explanation:

Step1: Recall limit definition

The limit $\lim_{x\rightarrow a}f(x)$ exists if and only if $\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a^{+}}f(x)$.

Step2: Analyze $\lim_{x\rightarrow 1^{-}}f(x)$

As $x$ approaches $1$ from the left - hand side, by looking at the graph, the function values approach a non - zero value.

Step3: Analyze $\lim_{x\rightarrow 1^{+}}f(x)$

As $x$ approaches $1$ from the right - hand side, the function values approach $0$. Since $\lim_{x\rightarrow 1^{-}}f(x)\neq\lim_{x\rightarrow 1^{+}}f(x)$, $\lim_{x\rightarrow 1}f(x)$ does not exist (DNE).

Step4: Analyze $\lim_{x\rightarrow 4^{-}}f(x)$ and $\lim_{x\rightarrow 4^{+}}f(x)$

As $x$ approaches $4$ from both the left - hand side and the right - hand side, the function values approach $0$. So $\lim_{x\rightarrow 4}f(x) = 0$. Also, from the graph, the value of the function at $x = 4$, $f(4)=0$.

Answer:

1. $\lim_{x\rightarrow 1}f(x)=DNE$
2. $\lim_{x\rightarrow 1^{+}}f(x)=0$
3. $\lim_{x\rightarrow 1}f(x)=DNE$
4. $\lim_{x\rightarrow 4}f(x)=0$
5. $f(4)=0$