Does exist for the following

Here is the graph of f. We have used point style for the graph otherwise the software would have incorrectly drawn in a vertical line connecting the two parts of the function.

We know that = 1 since f equals x for x>1.

We also know that = -3 since when x is less than or equal to 1.

Because the left and right limits are not equal, the limit of f as x approaches 1 does not exist.

In the rest of this note we will go over how to use the definition to show that the limit is not 1 as x approaches 1.

We will use and show that for all positive values of delta there are x values which are delta close to a = 1 whose corresponding y values are not epsilon close to L = 1.

In the graph below and

We can see that there are x values within delta = 0.25 of a = 1 (inside the green lines) whose y values do not fall within epsilon = .5 of L =1 (inside the red lines).

We continue with and use

Again there are x values within delta units of a = 1 whose y values are outside of the red target lines. We can continue with even smaller values of delta but each time there will always be x values delta close to a = 1 (to the left of a = 1) whose y values will not be within 0.5 units of L =1.

Thus, the limit as x approaches 1 cannot be 1.