Question

1. $lim_{x

ightarrow2^{-}}g(x)=2$ help (limits)

1. $lim_{x

ightarrow2^{+}}g(x)= - 2$

1. $lim_{x

ightarrow2}g(x)=dne$

1. $lim_{x

ightarrow0}g(x)= - 2$

1. $g(2)=1$

Explanation:

Step1: Analyze left - hand limit as $x\to2$

The left - hand limit $\lim_{x\to2^{-}}g(x)$ is given as 2. This means as $x$ approaches 2 from the left side, the function $g(x)$ approaches 2.

Step2: Analyze right - hand limit as $x\to2$

The right - hand limit $\lim_{x\to2^{+}}g(x)$ is given as - 2. This means as $x$ approaches 2 from the right side, the function $g(x)$ approaches - 2.

Step3: Determine limit as $x\to2$

Since $\lim_{x\to2^{-}}g(x)=2$ and $\lim_{x\to2^{+}}g(x)= - 2$, and $2
eq - 2$, by the definition of the limit (a limit $\lim_{x\to a}g(x)$ exists if and only if $\lim_{x\to a^{-}}g(x)=\lim_{x\to a^{+}}g(x)$), we have $\lim_{x\to2}g(x)$ does not exist (DNE).

Step4: Analyze limit as $x\to0$

The limit $\lim_{x\to0}g(x)= - 2$ as given.

Step5: Analyze function value at $x = 2$

The value of the function $g(2)$ is given as 1.

Answer:

1. $\lim_{x\to2^{-}}g(x)=2$
2. $\lim_{x\to2^{+}}g(x)= - 2$
3. $\lim_{x\to2}g(x)=DNE$
4. $\lim_{x\to0}g(x)= - 2$
5. $g(2)=1$