Question

for the function g(x) graphed here, find the following limits or state that they do not exist.
a. $\lim \limits_{x\to 2} g(x)$
b. $\lim \limits_{x\to 4} g(x)$
c. $\lim \limits_{x\to 6} g(x)$
d. $\lim \limits_{x\to 4.5} g(x)$
...
a. what is $\lim \limits_{x\to 2} g(x)$? choose the correct answer below and, if necessary, fill in the answer box to your choice.
a. $\lim \limits_{x\to 2} g(x) = \square$
b. $\lim \limits_{x\to 2} g(x)$ does not exist

Explanation:

Part a: $\lim_{x\to 2} g(x)$

Step1: Recall limit definition

For $\lim_{x\to a} g(x)$ to exist, the left - hand limit ($\lim_{x\to a^{-}} g(x)$) and right - hand limit ($\lim_{x\to a^{+}} g(x)$) must be equal.

Step2: Analyze left - hand limit at $x = 2$

As $x$ approaches $2$ from the left (values less than $2$), we look at the graph. The left - hand limit $\lim_{x\to 2^{-}} g(x)$ is the $y$ - value the graph approaches as $x$ gets closer to $2$ from the left. From the graph, as $x\to 2^{-}$, $g(x)\to 4$.

Step3: Analyze right - hand limit at $x = 2$

As $x$ approaches $2$ from the right (values greater than $2$), the right - hand limit $\lim_{x\to 2^{+}} g(x)$ is the $y$ - value the graph approaches as $x$ gets closer to $2$ from the right. From the graph, as $x\to 2^{+}$, $g(x)\to 4$.

Step4: Compare left and right limits

Since $\lim_{x\to 2^{-}} g(x)=4$ and $\lim_{x\to 2^{+}} g(x)=4$, by the definition of the limit, $\lim_{x\to 2} g(x) = 4$.

Part b: $\lim_{x\to 4} g(x)$

Step1: Recall limit definition

We need to check the left - hand limit ($\lim_{x\to 4^{-}} g(x)$) and right - hand limit ($\lim_{x\to 4^{+}} g(x)$).

Step2: Analyze left - hand limit at $x = 4$

As $x$ approaches $4$ from the left, looking at the graph, the $y$ - value the graph approaches is $4$. So $\lim_{x\to 4^{-}} g(x)=4$.

Step3: Analyze right - hand limit at $x = 4$

As $x$ approaches $4$ from the right, the $y$ - value the graph approaches is also $4$. So $\lim_{x\to 4^{+}} g(x)=4$.

Step4: Compare left and right limits

Since $\lim_{x\to 4^{-}} g(x)=\lim_{x\to 4^{+}} g(x) = 4$, $\lim_{x\to 4} g(x)=4$.

Part c: $\lim_{x\to 6} g(x)$

Step1: Recall limit definition

Check $\lim_{x\to 6^{-}} g(x)$ and $\lim_{x\to 6^{+}} g(x)$.

Step2: Analyze left - hand limit at $x = 6$

As $x$ approaches $6$ from the left, the graph is a line. We can see that as $x\to 6^{-}$, the $y$ - value approaches $- 2$ (assuming the grid and the line's slope, from the left - hand side of $x = 6$, the line is decreasing and approaches $y=-2$).

Step3: Analyze right - hand limit at $x = 6$

As $x$ approaches $6$ from the right, the graph is a line. As $x\to 6^{+}$, the $y$ - value also approaches $- 2$ (the line on the right - hand side of $x = 6$ is also approaching $y = - 2$).

Step4: Compare left and right limits

Since $\lim_{x\to 6^{-}} g(x)=\lim_{x\to 6^{+}} g(x)=-2$, $\lim_{x\to 6} g(x)=-2$.

Part d: $\lim_{x\to 4.5} g(x)$

Answer:

s:
a. $\lim_{x\to 2} g(x)=\boldsymbol{4}$ (Option A with value $4$)
b. $\lim_{x\to 4} g(x)=\boldsymbol{4}$
c. $\lim_{x\to 6} g(x)=\boldsymbol{-2}$
d. $\lim_{x\to 4.5} g(x)=\boldsymbol{3}$