use the given graph of the function g to find the following limits: 1. $lim_{x\rightarrow2^{+}}g(x)=2$ help…

Explanation:

Step1: Recall limit definition

The limit as $x$ approaches $a$ from the right, $\lim_{x\rightarrow a^{+}}g(x)$, is the value $y$ that the function approaches as $x$ gets closer to $a$ through values greater than $a$. The limit as $x$ approaches $a$ from the left, $\lim_{x\rightarrow a^{-}}g(x)$, is the value $y$ that the function approaches as $x$ gets closer to $a$ through values less than $a$. The two - sided limit $\lim_{x\rightarrow a}g(x)$ exists if and only if $\lim_{x\rightarrow a^{-}}g(x)=\lim_{x\rightarrow a^{+}}g(x)$.

Step2: Analyze $\lim_{x\rightarrow 2^{+}}g(x)$

Looking at the graph, as $x$ approaches $2$ from the right (values of $x>2$), the function $g(x)$ approaches $2$. So $\lim_{x\rightarrow 2^{+}}g(x) = 2$.

Step3: Analyze $\lim_{x\rightarrow 2^{-}}g(x)$

As $x$ approaches $2$ from the left (values of $x < 2$), the function $g(x)$ approaches $- 2$. So $\lim_{x\rightarrow 2^{-}}g(x)=-2$.

Step4: Analyze $\lim_{x\rightarrow 2}g(x)$

Since $\lim_{x\rightarrow 2^{-}}g(x)=-2$ and $\lim_{x\rightarrow 2^{+}}g(x)=2$, and $-2\neq2$, the two - sided limit $\lim_{x\rightarrow 2}g(x)$ does not exist (dne).

Step5: Analyze $\lim_{x\rightarrow 0}g(x)$

As $x$ approaches $0$ from both the left and the right, the function $g(x)$ approaches $-2$. So $\lim_{x\rightarrow 0}g(x)=-2$.

Step6: Analyze $g(2)$

The value of the function $g(x)$ at $x = 2$ is given by the solid - dot on the graph at $x = 2$. From the graph, $g(2)=1$.

Answer:

1. $\lim_{x\rightarrow 2^{+}}g(x)=2$
2. $\lim_{x\rightarrow 2^{-}}g(x)=-2$
3. $\lim_{x\rightarrow 2}g(x)=\text{dne}$
4. $\lim_{x\rightarrow 0}g(x)=-2$
5. $g(2)=1$