Question

for the function g(x) graphed here, find the following limits or state that they do not exist.
a. $limlimits_{x \to 2} g(x)$
b. $limlimits_{x \to 4} g(x)$
c. $limlimits_{x \to 6} g(x)$
d. $limlimits_{x \to 6.5} g(x)$

Explanation:

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

Step1: Analyze left and right limits

To find $\lim_{x \to 2} g(x)$, we check the left - hand limit (as $x$ approaches 2 from values less than 2) and the right - hand limit (as $x$ approaches 2 from values greater than 2). From the graph, as $x$ approaches 2 from the left, the function values approach 2. As $x$ approaches 2 from the right, the function values also approach 2 (since the line approaching $x = 2$ from the right is moving towards the $y$-value of 2).

Step2: Determine the limit

Since the left - hand limit $\lim_{x\to 2^{-}}g(x)=2$ and the right - hand limit $\lim_{x\to 2^{+}}g(x)=2$, by the definition of the limit (if $\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)=L$, then $\lim_{x\to a}f(x)=L$), we have $\lim_{x \to 2} g(x) = 2$.

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

Step1: Analyze left and right limits

For $\lim_{x \to 4} g(x)$, we look at the left - hand limit (as $x$ approaches 4 from the left) and the right - hand limit (as $x$ approaches 4 from the right). From the graph, as $x$ approaches 4 from the left, the function values approach 2, and as $x$ approaches 4 from the right, the function values also approach 2.

Step2: Determine the limit

Since $\lim_{x\to 4^{-}}g(x)=2$ and $\lim_{x\to 4^{+}}g(x)=2$, we conclude that $\lim_{x \to 4} g(x)=2$.

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

Step1: Analyze left and right limits

To find $\lim_{x \to 6} g(x)$, we check the left - hand limit and the right - hand limit. As $x$ approaches 6 from the left, the function values approach 0 (from the graph, the line approaching $x = 6$ from the left is moving towards $y = 0$). As $x$ approaches 6 from the right, the function values approach 2 (the line approaching $x = 6$ from the right is moving towards $y = 2$).

Step2: Determine the limit

Since $\lim_{x\to 6^{-}}g(x)=0$ and $\lim_{x\to 6^{+}}g(x)=2$, and $0
eq2$, the limit $\lim_{x \to 6} g(x)$ does not exist.

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

Answer:

s:
a. $\boldsymbol{2}$
b. $\boldsymbol{2}$
c. Does not exist
d. $\boldsymbol{1.5}$ (or $\boldsymbol{\frac{3}{2}}$)