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.

Does it show algebra steps or a proof?

The page shows the symbolic result, the two one-sided readings and the values used to check them. It does not provide a step-by-step algebra derivation. Sampled values are evidence rather than proof; a result supported only by samples is labelled an estimate with an approximately sign.

How do I enter a limit written as h approaches 0?

For the limit of sin(h)/h as h approaches 0, enter sin(h)/h in Expression, h in Variable and 0 in Point, then choose Two-sided. The result is 1. Use a single letter for the moving variable; type the expression alone rather than the full limit statement.

Can I see a limit graph and table of values?

Sample graph plots the finite values listed in The numbers behind it, with the input on the horizontal axis and the expression value on the vertical axis. The left and right samples stay separate, and a dashed line marks a finite approach point. The axes are linear and dots are not joined, so the graph shows only the sampled evidence. Enter a formula to generate it; graph uploads and precomputed table inputs are not supported.

Can I use the limit calculator free online?

Yes. Use it in a current browser on a computer, tablet or phone without an account or installation. The expression is read in the page, and the symbolic calculation runs in a background worker using a library served by this site. Your last inputs are kept in this browser.

Limit Calculator

Evaluate a calculus limit with one variable approaching a point or infinity, and compare the two one-sided results. Enter the expression, variable and point in the controls below. The symbolic answer is checked against six values approaching a finite point from each side; a conflict is labelled Unresolved.

expressionx → 0

sin(x)/x

variable   x
point      0
direction  two-sided

limitworking

Working

from the left    ≈ 1  (sampled)
from the right   ≈ 1  (sampled)

The symbolic engine is still running. The sampled values are already below.

Expression

Variable

Limit: Working
Point
Direction
Result: Working

Working. The symbolic engine is still running. The sampled values are already below.

sin(x)/x at 0two-sided
1
(1 - cos(x))/x² at 0two-sided
1/2
1/x at 0left to -∞, right to +∞
no two-sided limit

Accuracy. 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.

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.

Sample graph

Dots show the same finite values as the tables on linear axes. The left and right approaches stay separate; no curve is inferred between samples.

● from the left: 6 finite samples● from the right: 6 finite samples

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.
Does it show algebra steps or a proof?: The page shows the symbolic result, the two one-sided readings and the values used to check them. It does not provide a step-by-step algebra derivation. Sampled values are evidence rather than proof; a result supported only by samples is labelled an estimate with an approximately sign.
How do I enter a limit written as h approaches 0?: For the limit of sin(h)/h as h approaches 0, enter sin(h)/h in Expression, h in Variable and 0 in Point, then choose Two-sided. The result is 1. Use a single letter for the moving variable; type the expression alone rather than the full limit statement.
Can I see a limit graph and table of values?: Sample graph plots the finite values listed in The numbers behind it, with the input on the horizontal axis and the expression value on the vertical axis. The left and right samples stay separate, and a dashed line marks a finite approach point. The axes are linear and dots are not joined, so the graph shows only the sampled evidence. Enter a formula to generate it; graph uploads and precomputed table inputs are not supported.
Can I use the limit calculator free online?: Yes. Use it in a current browser on a computer, tablet or phone without an account or installation. The expression is read in the page, and the symbolic calculation runs in a background worker using a library served by this site. Your last inputs are kept in this browser.

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.