Limits & Continuity Explorer
Enter a function and a point to evaluate the limit from both sides. The tool builds a table of approaching values, graphs the function, and classifies any discontinuity.
Input
Limit Results
Graph
Values Approaching x = 0
From the Left (x < 0)
| x | f(x) |
|---|---|
| -0.10000000 | 0.99833417 |
| -0.010000000 | 0.99998333 |
| -0.0010000000 | 0.99999983 |
| -0.00010000000 | 1.0000000 |
| -0.000010000000 | 1.0000000 |
| -0.0000010000000 | 1.0000000 |
From the Right (x > 0)
| x | f(x) |
|---|---|
| 0.10000000 | 0.99833417 |
| 0.010000000 | 0.99998333 |
| 0.0010000000 | 0.99999983 |
| 0.00010000000 | 1.0000000 |
| 0.000010000000 | 1.0000000 |
| 0.0000010000000 | 1.0000000 |
Step-by-Step
1. Evaluate the left-hand limit
2. Evaluate the right-hand limit
3. Two-sided limit
4. Classification
Removable Discontinuity
Open this tool with your own values
This tool reads its inputs from the page address, so a link can open it with the values from a problem already entered. Include every value the problem needs, because one left out keeps the tool's default. Use plain numbers in the stated units.
- f
- function of x, such as (x^2-9)/(x-3)
- pt
- x value the limit approaches
Example link
Reference Guide
Left and Right Limits
The left-hand limit approaches from values less than a; the right-hand limit from values greater than a.
Two-Sided Limit
The two-sided limit exists only when both one-sided limits exist and are equal.
Continuity
A function is continuous at x = a when the limit exists, the function is defined there, and they are equal.
Types of Discontinuity
Removable means the limit exists but the function is undefined or has a different value there. Jump means the left and right limits differ. Infinite means the function blows up.
