Two-Sided Limit From a Graph

1) What a two-sided limit means

A two-sided limit asks for the value the graph approaches as (x) gets close to a point from both sides. You do not need the function value at that point; you only need the behavior of the graph near it.

2) Read the graph from left and right

• Move toward the target (x)-value from the left and note the (y)-value the graph approaches.
• Move toward the same (x)-value from the right and do the same.
• If both sides approach the same number, that number is the two-sided limit.
• If the left and right approaches are different, the two-sided limit does not exist.

3) Watch for common graph features

Open circles, jumps, and holes can make the function value differ from the limit. A solid dot at the point does not automatically give the limit. The limit depends on nearby values, not the plotted value itself.

4) How to answer

Write the limit as the number approached, or write that it does not exist when the two sides disagree.

5) Quick check

Confirm that the left-hand and right-hand approaches match exactly. If they do, your answer is consistent; if not, the limit is undefined or does not exist.