This calculator combines substitution checks, one-sided sampling, numeric LHopital, and a graph around the approach point. Copy a share link, LaTeX summary, or CSV table for class handouts.
Beyond polynomials and trigonometric functions, you can call abs, sgn, sqrt, and constants such as pi, +inf, and -inf.
Inputs & options
Graph near a
Result & samples
How it's calculated
How to use this calculator effectively
Use this explorer when you want a numeric picture of a limit together with one-sided samples, a graph, and short steps that explain why the result is finite, divergent, or one-sided only.
Choose the approach point and side
Enter the expression, then set the point a and choose whether you want the two-sided limit or only the left or right side. One-sided mode is the right tool for roots, absolute values, and functions that are undefined on one side.
How to read the table and graph
The table samples values closer and closer to the approach point, while the graph shows whether both sides settle toward the same height. Read them together: a stable number in the table should agree with the graph and the step summary.
What this calculator is good for
It is useful for classroom checking, spotting removable discontinuities, and testing whether a candidate algebraic simplification matches the local behaviour near a point.
Common mistakes to avoid
- Typing a piecewise idea as one expression without checking whether the relevant side is defined.
- Reading a rapidly growing value as convergence just because the last few samples look similar.
- Ignoring the side selector when the left and right behaviour obviously differ.
When to use another page instead
Use the graphing calculator for wider function exploration, and use symbolic algebra tools when you need exact manipulation before checking the local numeric behaviour here.
FAQ
How does the numeric LHopital routine work?
When both numerator and denominator approach 0 or ∞, the calculator estimates derivatives with Richardson extrapolation. It applies LHopital up to six times and logs each attempt.
How are one-sided limits visualized?
Samples at 10-1 to 10-6 from the approach point appear in both the table and graph. If left and right values differ, the result is marked as one-sided only.
Why can the graph and the table matter more than the final line?
A limit can fail even when several nearby sample values look close together. The table, graph, and side-specific samples help you see oscillation, divergence, or left/right disagreement before trusting the summary line.
When should I use left-only or right-only mode?
Use one-sided mode when the function is defined on only one side of the point, or when the two sides clearly behave differently. Typical examples are sqrt(x) at 0 and rational functions with vertical asymptotes.
Can this replace a symbolic proof?
No. This page is a numeric and visual check. It is very useful for building intuition and catching mistakes, but a formal proof still needs algebra, identities, or theorem-based reasoning outside the numeric sample.
How to interpret the output
Finite vs divergent behaviour
A finite limit should show both sides settling toward the same number. Divergence usually appears as one side growing without bound, the two sides separating, or the samples refusing to stabilise as the step size shrinks.
Removable vs non-removable discontinuity
If the function value at a is missing or different but the nearby samples agree, you are looking at a removable discontinuity. If the sides disagree or blow up, the discontinuity is not removable.
How to use this in teaching
Start with a familiar form such as sin(x)/x or (1-cos(x))/x^2, then compare the numeric evidence with the algebraic simplification students already know. This makes the role of one-sided checks much clearer.
What to verify before sharing a result
Check the expression syntax, the approach point, and the selected side. Then confirm that the graph, table, and step summary all support the same conclusion before copying the URL or LaTeX.
Next step after the numeric check
Once the local behaviour looks correct, move to the algebraic proof or class notes that justify it formally. This calculator is best used to support that reasoning, not to replace it.