Question

Use the given graph to determine the limit, if it exists.

Answer 1

The limit is the value of the function when you approach the closest possible  to the x value without getting to that x value.

The closed point (2,1) is not used to calculate the limit. The closed point (2,1) represents the value of the function at x = 1, but the limit is not the value of the function.

Use the horizontal line that comes from the left and ends at the open point (2,3) to find the limit of f(x) when x approaches to 2 from the left.

That limit is 3, because you can get as close as you want to 3 when you approach x = 2 from the left.

Use the horizontal line that starts at the open point (2-2) and goes to the right, to find the limit of f(x) when x approached to 2 from the right.

That limit is -2 because you can get as close to -2 when you approach x = 2 from the right.

It is important that you know that the definition of limits do not include the value of the function at the point where you want to state the limit.

This result, limit of f(x) when x approaches 2 from the left  = 3  and limit of f(x) when x approaches 2 from the right = -2, means that the limit of the function when x approaches 2 does not exist.

The limit exists if and only if the two lateral limits exists and are equal.