Limit Calculator
Type an expression, the variable, the point it approaches and which side you want. The symbolic answer is computed by a computer algebra library running in a sandboxed worker, and it is checked against the expression itself at six points walking in toward that point from each side, or eight walking out toward infinity. The two methods cover each other: the algebra gives an exact constant such as 1/2 rather than 0.4999999, and the numbers catch the cases where the library answers confidently and wrongly. The one-sided results sit next to the two-sided one, because a limit can exist from one side and not from the other, and because that is what tells you whether there is a two-sided limit at all.
sin(x)/x variable x point 0 direction two-sided
Working
from the left ≈ 1 (sampled)
from the right ≈ 1 (sampled)
The symbolic engine is still running. The sampled values are already below.
- Limit
- Point
- Direction
- Result
Working. The symbolic engine is still running. The sampled values are already below.
The numbers behind it
Every reading above is checked against the expression itself, one decade at a time. These are those values, so nothing is hidden behind the word approximately.
| x, from the left | value |
|---|---|
| -0.1 | 0.99833417 |
| -0.01 | 0.99998333 |
| -0.001 | 0.99999983 |
| -0.0001 | 1 |
| -0.00001 | 1 |
| -0.000001 | 1 |
| x, from the right | value |
|---|---|
| 0.1 | 0.99833417 |
| 0.01 | 0.99998333 |
| 0.001 | 0.99999983 |
| 0.0001 | 1 |
| 0.00001 | 1 |
| 0.000001 | 1 |
How this is worked out
The symbolic answer comes from nerdamer 1.1.13, running in a worker of its own with a 2 second clock beside it: some expressions send it a long way, and the page stops it rather than freezing with it. The one-sided questions are asked as their own limits, with the variable replaced by the point plus or minus 1/t as t goes to infinity, which walks onto exactly one side of the point. At a point written with pi or e that substitution is skipped, because the library swaps in a rational approximation of the constant and answers for that number instead; those sides are read from the samples and marked as sampled.
The samples are taken here on the page: 6 points walking in toward a finite point from each side, or out to ten to the 8 for a point at infinity. They can contradict a symbolic answer and take it away, and on their own they only ever give an estimate or a description of what the values are doing, never an exact limit. When the two methods disagree the page prints both and calls the result unresolved, because a sampled value can be distorted by cancellation and a symbolic answer can come back without being checked.
What you can type
Numbers, single letters for variables, the constants pi and e, the signs + - * / ^ and brackets, and these functions: sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs, log. ln is read as log, which is the natural logarithm here, and arcsin, arccos and arctan are read as asin, acos and atan. Multiplication can be left out where it is obvious, so 2x and 3 sin(x) read the way they are written. The point can be a number, a fraction such as 1/2, pi/2, e, or infinity and -infinity. Expressions are read up to 500 characters.
Common questions
- What does "no two-sided limit" mean here?
- It means the two one-sided results disagree, so the ordinary limit does not exist. For 1/x at 0 the values fall without bound on the left and rise without bound on the right, and the page says so instead of printing a number. It is a different result from a limit that diverges: 1/x squared at 0 rises on both sides, which the page labels Diverges and reports as +infinity in the extended sense, with no finite limit.
- Why does it sometimes say the result is unresolved?
- Because the symbolic engine and the sampled values disagreed, and the page will not choose between them for you. A sampled value can be distorted by floating point cancellation near the point, and a symbolic answer can come back without being verified. When they conflict, both readings are shown and neither is printed as the limit.
- Can I take a limit from one side only?
- Yes. Choose From the left or From the right and the answer is for that side. The other side is still worked out and reported underneath, because a one-sided limit can exist where the two-sided one does not. Where the two-sided limit does exist, both one-sided limits equal it.
- Does it handle limits at infinity?
- Yes. Type infinity or -infinity as the point. There is only one direction to approach from, so the page shows a single approach labelled from below or from above, and it samples the expression out to ten to the eighth to check the symbolic answer.
- What can I type into the expression?
- Numbers, single letters for variables, the constants pi and e, the signs plus, minus, times, divide and caret, brackets, and the functions sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs and log. ln is read as log, which is the natural logarithm here. Multiplication can be left out where it is obvious, so 2x and 3 sin(x) read the way they are written. The expression is read up to 500 characters and powers up to 64.
- Is the answer a proof?
- The exact answers are the symbolic engine's, and the sampled values are evidence rather than proof: they can support or contradict an answer, and the page never turns a settling set of samples into an exact result. When only the samples settle, the answer is labelled an estimate and shown with an approximately sign.
Limits are evaluated symbolically in a sandboxed worker, so an exact constant such as 1/2 stays 1/2 rather than becoming 0.4999999. The one-sided limits are computed separately and the page reports that no two-sided limit exists when they disagree. Existence is never inferred from a sampled graph, and an expression the parser cannot handle is labelled unsupported instead of answered.