+25XP

MathematicsCalculus

Limit calculator with steps

Published 8 Nov 20269 min read

This limit calculator finds the limit of a function as x approaches a number, or as x goes to infinity, from both sides or from one side. It works the limit out with the right method, such as substitution, factor and cancel, L'Hôpital's rule or comparing the highest powers, and shows every step. A table of values and a graph show the function closing in, so you can see why the answer is right.

It opens on the limit of sin x over x as x approaches 0. Putting 0 straight in gives 0/0, which says nothing, but the calculator shows that the limit is exactly 1.

Try it

Limit Calculator

Evaluate limits and watch the function approach its value

Go to its page

How to use the limit calculator

Type the function in the f(x) box. Typing pi, sqrt or infty makes the symbol, and / builds a fraction.
Under Approaching, choose A point, +∞ or −∞.
For a point, type the value in x approaches. It can be a number or a value like pi/2.
Choose the side: Both for an ordinary limit, or Left or Right for a one-sided limit.
Read the answer and the working. The readouts give the limit from the left and from the right, and the value of f at the point itself, which is often undefined.

Under the answer, a table lists f(x) for x getting closer to the point from each side, by 0.1, 0.01, 0.001 and 0.0001. The graph steps points in from each side towards the limit. If no rule settles a limit, the calculator estimates it from sampling and says so. For practice, Practice and help sets random limits with hints and worked solutions.

How limits are worked out

The calculator tries the methods below and names the one it used. Every row can be typed in to check.

Method	When it works	Type this	Limit
Direct substitution	The function is continuous at the point	x^2+3x-1, x approaches 2	9
Factor and cancel	Substitution gives 0/0 and a factor cancels	(x^2+x-6)/(x-2), x approaches 2	5
L'Hôpital's rule	Substitution gives 0/0 or ∞/∞	(e^x-1)/x, x approaches 0	1
Highest powers	A fraction of polynomials as x goes to ∞	(3x^2-5x)/(2x^2+7), +∞	3/2
One-sided limits	The two sides behave differently	\|x\|/x, x approaches 0	Does not exist
Vertical asymptote	The bottom goes to 0 but the top does not	1/x^2, x approaches 0	∞
Take logs	A power where both parts change, like 1 to the power ∞	(1+1/x)^x, +∞	e

Worked examples you can check in the calculator

Example 1: factor and cancel

Find the limit of (x² − 9)/(x − 3) as x approaches 3. Putting in 3 gives 0/0. That does not mean the limit is 0 or undefined: it means the top and bottom share a factor. Factor the top and cancel:

\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} = \lim_{x \to 3} (x + 3) = 6

The calculator gives 6 and names the method "Factor and cancel". Its readouts show that f(3) itself is undefined, even though the limit is 6. A limit is about where the function is heading, not what happens at the point.

Example 2: L'Hôpital's rule twice

Find the limit of (1 − cos x)/x² as x approaches 0. Substituting gives 0/0, so differentiate the top and the bottom separately. That gives sin x / 2x, which is still 0/0 at 0, so do it again:

\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \lim_{x \to 0} \frac{\sin x}{2x} = \lim_{x \to 0} \frac{\cos x}{2} = \frac{1}{2}

The calculator's working shows the rule applied twice before it reaches 1/2. L'Hôpital's rule only applies to 0/0 or ∞/∞ forms, and it differentiates the top and bottom on their own, not as a quotient.

Example 3: a limit you can see closing in

The opening example, sin x / x as x approaches 0, cannot be factored, but a table makes the answer clear:

x	sin x / x
0.1	0.99833417
0.01	0.99998333
0.001	0.99999983
0.0001	0.99999999...

The values get closer and closer to 1, and because sin x / x is even, the negative side gives the same values. The calculator shows a table like this for every limit at a point, with both sides side by side.

y = \frac{\sin\left( x\right)}{ x}

Open in graph

Drag to pan, scroll to zoom. x from -10 to 10, y from -0.46 to 1.24.

Here x is the input in radians and y is sin x / x. The graph has a gap at x = 0, where the function is undefined, but both sides head for the height y = 1. Try a limit in the course now, then press Mark:

Press Mark to have your answer marked in the Calculus course.

Limits at infinity

Choose +∞ or −∞ to see what happens as x grows without end. For a fraction of two polynomials, only the highest power on the top and on the bottom matters:

Highest powers	Limit as x goes to ∞	Example	Answer
Same on top and bottom	The ratio of their coefficients	(3x² − 5x)/(2x² + 7)	3/2
Higher on the bottom	0	(4x + 1)/(x² − 2)	0
Higher on the top	∞ or −∞	x³/(5x² + 1)	∞

A finite limit at infinity is a horizontal asymptote of the graph. So y = (3x² − 5x)/(2x² + 7) levels off at y = 3/2 as x gets large. Exponentials beat any power: the limit of e^(−x) as x goes to ∞ is 0.

One-sided limits and when a limit does not exist

A limit at a point exists only when the limit from the left and the limit from the right are the same number. Set the side to Left or Right to find each one, or leave it on Both and read both from the readouts.

Example 4: two different sides

Find the limit of 1/x as x approaches 0. From the right, x is a tiny positive number, so 1/x is huge and positive. From the left it is huge and negative:

\lim_{x \to 0^{-}} \frac{1}{x} = -\infty \quad \lim_{x \to 0^{+}} \frac{1}{x} = \infty

The sides differ, so the calculator says the limit does not exist, with the note "The left and right limits are different". Set the side to Right and it gives ∞. Compare 1/x² at 0: both sides go to +∞, so the calculator gives ∞ as the limit. The same idea explains |x|/x at 0, which is −1 from the left and 1 from the right, so it does not exist either.

A one-sided limit can also be undefined when the function has no values on that side. The square root of x has no real values for x below 0, so its limit from the left at 0 is undefined.

Indeterminate forms

Some results of substituting tell you nothing on their own. The calculator calls these "a form that says nothing on its own" and moves to another method.

Form	Example	Usual method
0/0	(x² − 9)/(x − 3) at 3	Factor and cancel, or L'Hôpital's rule
∞/∞	(3x² − 5x)/(2x² + 7) at ∞	Highest powers, or L'Hôpital's rule
1 to the power ∞	(1 + 1/x)^x at ∞	Take logs
0 × ∞ and ∞ − ∞	x ln x at 0 from the right	Rewrite as a fraction first

The limit of (1 + 1/x)^x as x goes to ∞ is e, about 2.71828, one of the most famous limits in maths. The calculator reaches it by taking logs, rewriting as a fraction and using L'Hôpital's rule.

Common mistakes

Saying 0/0 means the limit is 0 or undefined. It means you need another method, such as factoring.
Using L'Hôpital's rule when the form is not 0/0 or ∞/∞. Check the form first, every time you apply it.
Using the quotient rule in L'Hôpital's rule. Differentiate the top and the bottom separately.
Ignoring the two sides. Check both sides when the bottom of a fraction goes to 0. If they differ, the limit does not exist.
Mixing up the limit with the value. f(a) can be undefined while the limit exists, or different from it.
Working in degrees. sin x / x approaches 1 only when x is in radians.

Practise limits in a short timed paper

Limits open AP Calculus AB and AP Calculus BC, and they underpin differentiation in A-level Maths: Calculus in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics. Limits lead straight to derivatives, which the derivative calculator guide covers, and the Calculus course starts with them.

Common questions

What is a limit in calculus?

The value a function gets closer and closer to as x approaches a point or grows without end. The function does not have to reach it, or even be defined at the point.

How do you find a limit?

Try substituting the value first. If you get a number, that is the limit. If you get 0/0 or ∞/∞, factor and cancel, compare the highest powers, or use L'Hôpital's rule.

When does a limit not exist?

When the left and right limits are different, as for 1/x or |x|/x at 0, or when the function keeps oscillating without settling. The calculator says "does not exist" and gives the reason.

What is a one-sided limit?

A limit from one side only: x approaching from below (the left, written 0⁻) or from above (the right, written 0⁺). Set the side to Left or Right in the calculator.

What is L'Hôpital's rule?

If substituting gives 0/0 or ∞/∞, the limit of f(x)/g(x) equals the limit of f'(x)/g'(x), when that exists. You may need to apply it more than once.

Can the calculator do multivariable limits?

No. It finds limits of functions of one variable. If the function has a second letter, it asks you for a number for it.

Revision cards

Say the answer first, then flip the card to check it.

Where to go next

Solve it
Limit Calculator
Evaluate limits and watch the function approach its value
Open the tool
Learn it
Calculus
Differentiation, integration and the mathematics that drives physics and engineering
Start the course
Test it
AP Calculus AB
A new exam-style paper every day, with instant marking and a mark scheme
Start today's paper