Limit Calculator
Estimate one- and two-sided limits numerically.
Table of Contents
How to Use
- Enter a function of one variable using x (e.g., sin(x)/x or (x^2-1)/(x-1)).
- Pick the variable name, the value x approaches, and whether you want a two-sided or one-sided limit.
- Run the calculation to sample points on each side of the approach value.
- Compare the left and right estimates to decide if the limit converges.
What does a limit describe?
A limit captures the value a function approaches as the input moves toward a specific point. If the left-hand and right-hand values match, the two-sided limit exists. If they differ or diverge, the limit does not exist.
- Two-sided limits require matching left- and right-hand behavior.
- One-sided limits only consider values from one direction.
- Removable discontinuities can still have matching limits even if the function is undefined at the point.
How this calculator estimates limits
- Sample function values at progressively smaller offsets from the approach point.
- Track left-hand and right-hand sequences separately to reveal asymmetry.
- Compare the latest samples to judge if values are converging.
- Report the estimated limit or highlight divergence when sides disagree.
Numerical sampling is helpful for quick intuition. For proofs or symbolic certainty, combine it with algebraic techniques.
Frequently Asked Questions
- Why do left and right limits matter?
- A two-sided limit only exists when both sides agree. If the left-hand and right-hand limits differ or one side diverges, the overall limit does not exist at that point.
- Can this handle infinite limits?
- The calculator reports divergence when sampled values grow without bound or do not agree. You can still spot vertical asymptotes by examining the left and right sample values.
- Is this a symbolic solver?
- No. The tool uses numerical sampling to approximate limits. For exact symbolic work, pair the result with algebraic simplification or analytic rules such as L'Hôpital's rule.