One-Sided Limits From a Graph

One-Sided Limits From a Graph

1. Identify the x-value and the side. A one-sided limit asks what value the function approaches as (x) gets close to a point from the left ((x \to a^-) or from the right ((x \to a^+)).

2. Read the graph near that point. Follow the curve or line segment toward the target x-value. Ignore the function’s value at the point itself if there is a filled or open dot unless the question asks for the function value.

3. Watch for common graph features.

   • An open circle often shows the approached value, not the actual function value.
   • A filled dot shows the value of the function at that x-value, which may be different from the limit.
   • A jump or vertical asymptote may mean the one-sided limit is different on each side or does not exist.

4. State the limit clearly. If the graph approaches a single y-value from the chosen side, that is the one-sided limit. If it increases or decreases without bound, write that it diverges or use (\infty) or (-\infty) as appropriate.

5. Check your answer. Make sure you used the correct side and that your y-value matches the height the graph is approaching, not just a marked point.

Quick check

Ask: “As I move along the graph from the requested side, what y-value do I get close to?”