About Limit Calculator
This limit calculator is free, needs no sign-up and runs in your browser, so the expressions you type are never uploaded. Enter an expression, say where x is heading, and it gives the limit with the working written out.
The steps say which method decided the answer: direct substitution, factoring and cancelling, a standard limit, L'Hopital's rule, comparing leading terms at infinity, or, as a last resort, a numeric estimate that is labelled as one. You can approach a number from the left, from the right or from both sides, or send x to infinity or minus infinity.
A graph shows the function near the point with the limit value drawn as a line, so you can see the function close in on the answer, or see why the two sides disagree.
How to use Limit Calculator
- Type the expression in x, for example (x^2-4)/(x-2), or use the key pad.
- In "x approaches" enter a number such as 2, a constant like pi/2, or inf or -inf.
- For a finite point pick Both sides, From the left or From the right. At infinity there is only one direction.
- Read the result card for the limit and the name of the method used.
- Open the steps to see the working, and use the graph to check the behaviour near the point.
- If the result is labelled a numeric estimate, treat it as a guide and look for an algebraic argument.
Features
- Limits at a number, at infinity and at minus infinity.
- Left-hand, right-hand and two-sided limits, with the two-sided limit reported as not existing when the sides disagree.
- Plus and minus infinity as results, for expressions that grow without bound.
- Steps that name the method: substitution, factor and cancel, standard limit, L'Hopital's rule or leading terms.
- A numeric cross-check, and a clear label when the answer is a numeric estimate.
- A graph around the point with the limit drawn as a horizontal line and the point marked.
- A live preview and a math key pad, so input is easy on a phone.
- Clear errors for input that cannot be read.
The methods, in the order they are tried
First the calculator tries to substitute the value. If that gives a number, that is the limit. If it gives 0/0 or infinity over infinity, it tries to factor and cancel, then standard limits, then L'Hopital's rule. At infinity it compares the highest powers.
A limit exists only when the left-hand and right-hand limits agree. If they differ, as for 1/x at 0, the two-sided limit does not exist.
lim (x->a) f(x) = L if f(x) gets as close to L as you like when x is near a
sin(x)/x -> 1 as x -> 0
(1 + 1/x)^x -> e as x -> infinity
L'Hopital: lim f/g = lim f'/g' for 0/0 or inf/infWorked examples
Limit of (x^2-4)/(x-2) as x approaches 2 gives 4
Factor the top as (x - 2)(x + 2) and cancel.
Limit of (1-cos(x))/x as x approaches 0 gives 0
L'Hopital's rule: sin(x)/1 at 0.
Limit of (3x^2+1)/(x^2-5) as x approaches infinity gives 3
Ratio of the leading coefficients.
Limit of 1/x as x approaches 0 from the right gives infinity
Large and positive for small positive x.
Limit of 1/x as x approaches 0 from the left gives -infinity
Large and negative for small negative x.
Good to know
What the calculator can and cannot do
It handles the limits met in a first calculus course: rational functions, roots, exponentials, logarithms, trigonometric functions and their products and quotients. It checks its symbolic answer against values of the function near the point, and if they disagree the numeric result is reported and labelled. At infinity it compares the fastest-growing terms, which covers exponentials with different bases such as 2^x/3^x, and it multiplies by the conjugate when square roots cancel, as in sqrt(x^2+x) - x. Sums of fractions such as 1/x - 1/sin(x) are combined over a common denominator, and powers such as x^x are handled with the logarithm trick, never by plugging in.
A numeric estimate is not a proof. For a limit that oscillates or needs a delicate argument, the calculator reports that the limit does not exist only when the one-sided values clearly disagree or the values oscillate. When the values simply do not settle it says the limit could not be determined. A numeric result is extrapolated from values near the point and matched to a simple exact value when one fits, and it is always labelled numeric, so check such cases by hand.
Privacy: how it works
- The limit is worked out by code running in your browser tab. Your expressions are not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- If direct substitution gives a number, you are done: that number is the limit for any continuous function.
- For 0/0, try factoring and cancelling before reaching for L'Hopital's rule.
- At infinity, divide the top and bottom by the highest power of x, or just compare the leading terms.
- Check the one-sided limits when the function has an absolute value, a hole or a vertical asymptote at the point.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your expressions are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your expressions are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I find a limit?
Try substituting the value first. If you get 0/0, factor and cancel, use a standard limit or apply L'Hopital's rule. At infinity, compare the highest powers. The calculator tries these in order and shows which one worked.
What does it mean when a limit does not exist?
The function does not settle on one value. The most common reason is that the left-hand and right-hand limits differ, as for 1/x at 0. It can also oscillate or grow without bound.
What is the difference between a one-sided and a two-sided limit?
A one-sided limit looks at only one direction, from the left or from the right. The two-sided limit exists only when both one-sided limits exist and are equal.
What is L'Hopital's rule?
If substituting gives 0/0 or infinity over infinity, the limit of f/g equals the limit of f'/g', provided that second limit exists. The calculator applies it, repeating if needed, and shows each round.
How do I take a limit at infinity?
Type inf or -inf in "x approaches". For rational functions the answer comes from the leading terms. For others the calculator uses L'Hopital's rule or comparison.
Can the answer be infinity?
Yes. If the function grows without bound, the result is shown as infinity or minus infinity. Strictly the limit does not exist as a number, and infinity describes how it fails to.
Why is the result labelled a numeric estimate?
When no exact method applied, the calculator reads the value from the function near the point. That is an estimate, so it is labelled one.
Popular Limit Calculator pages
- Limit of sin(x)/x as x approaches 0Step-by-step limit of sin(x)/x as x approaches 0, with the method named and a graph near the point.
- Limit of (1+1/x)^x as x approaches infinityStep-by-step limit of (1+1/x)^x as x approaches infinity, with the method named and a graph near the point.
- Limit of (e^x-1)/x as x approaches 0Step-by-step limit of (e^x-1)/x as x approaches 0, with the method named and a graph near the point.
- Limit of (1-cos(x))/x as x approaches 0Step-by-step limit of (1-cos(x))/x as x approaches 0, with the method named and a graph near the point.
- Limit of x*ln(x) as x approaches 0 from the rightStep-by-step limit of x*ln(x) as x approaches 0 from the right, with the method named and a graph near the point.
- Limit of (x^2-1)/(x-1) as x approaches 1Step-by-step limit of (x^2-1)/(x-1) as x approaches 1, with the method named and a graph near the point.
- Limit of (1-cos(x))/x^2 as x approaches 0Step-by-step limit of (1-cos(x))/x^2 as x approaches 0, with the method named and a graph near the point.
- Limit of (x^2+1)/(2x^2-3) as x approaches infinityStep-by-step limit of (x^2+1)/(2x^2-3) as x approaches infinity, with the method named and a graph near the point.
- Limit of sin(3x)/x as x approaches 0Step-by-step limit of sin(3x)/x as x approaches 0, with the method named and a graph near the point.
- Limit of tan(x)/x as x approaches 0Step-by-step limit of tan(x)/x as x approaches 0, with the method named and a graph near the point.
- Limit of ln(x)/x as x approaches infinityStep-by-step limit of ln(x)/x as x approaches infinity, with the method named and a graph near the point.
- Limit of x^2*e^(-x) as x approaches infinityStep-by-step limit of x^2*e^(-x) as x approaches infinity, with the method named and a graph near the point.
- Limit of 1/x as x approaches 0Step-by-step limit of 1/x as x approaches 0, with the method named and a graph near the point.
- Limit of 1/x as x approaches 0 from the rightStep-by-step limit of 1/x as x approaches 0 from the right, with the method named and a graph near the point.