Question

some function g is graphed below. fill in the blanks in the following sentences.
as x gets closer and closer to (but stays less than) 1, g(x) gets as close as we want to 3.
as x gets closer and closer to (but stays greater than) 1, g(x) gets as close as we want to 2.
as x gets closer and closer (but not equal) to 1, does g(x) get as close as we want to a single value? if such a value exists, enter it. if no such value exists, enter dne.

Explanation:

Step1: Analyze left - hand limit

As \(x\) approaches \(1\) from the left ( \(x<1\)), by observing the graph, the \(y\) - values of the function \(g(x)\) approach \(3\).

Step2: Analyze right - hand limit

As \(x\) approaches \(1\) from the right (\(x > 1\)), by observing the graph, the \(y\) - values of the function \(g(x)\) approach \(2\).

Step3: Determine overall limit

For the limit of \(g(x)\) as \(x\) approaches \(1\) to exist, the left - hand limit and the right - hand limit must be equal. Since \(\lim_{x\to1^{-}}g(x)=3\) and \(\lim_{x\to1^{+}}g(x)=2\), and \(3
eq2\), the limit of \(g(x)\) as \(x\) approaches \(1\) does not exist.

Answer:

DNE