The limit approaching from the left is different than the limit approaching from the right, thus, the limit does not exist, the correct option is the fourth one.
How to get the limit when the variable tends to 1?
So we want the limit when x tends to 1 (from both directions) of the function g(x), which is shown in the graph.
Now, remember that the limit never reaches the actual value x = 1, it studies how the function approaches to that value.
When we approach from the left (we approach x = 1 from values smaller than 1), we can see that the function tends to -2.
[tex]\lim_{x \to 1^-} g(x) = -2[/tex]
When we approach from the right, we get:
[tex]\lim_{x \to 1^+} g(x) = 1[/tex]
Now, the limit:
[tex]\lim_{x \to 1} g(x)[/tex]
Only exists if the two above ones are equal, which is not the case, so the limit does not exist.
So your answer is correct.
If you want to learn more about limits:
https://brainly.com/question/5313449
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