Question

use the given graph of the function g to find the following limits: 1. $lim_{x
ightarrow2^{+}}g(x)=2$ help (limits) 2. $lim_{x
ightarrow2^{-}}g(x)= - 2$ 3. $lim_{x
ightarrow2}g(x)= dne$ 4. $lim_{x
ightarrow0}g(x)= - 2$ 5. $g(2)=1$

Explanation:

Step1: Recall limit definition

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

Step2: Analyze $\lim_{x

ightarrow 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 $\lim_{x
ightarrow 2^{+}}g(x) = 2$.

Step3: Analyze $\lim_{x

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

Step4: Analyze $\lim_{x

ightarrow 2}g(x)$
Since $\lim_{x
ightarrow 2^{-}}g(x)=-2$ and $\lim_{x
ightarrow 2^{+}}g(x)=2$, and $-2
eq2$, the two - sided limit $\lim_{x
ightarrow 2}g(x)$ does not exist (dne).

Step5: Analyze $\lim_{x

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

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

ightarrow 2^{+}}g(x)=2$

1. $\lim_{x

ightarrow 2^{-}}g(x)=-2$

1. $\lim_{x

ightarrow 2}g(x)=\text{dne}$

1. $\lim_{x

ightarrow 0}g(x)=-2$

1. $g(2)=1$