Limit calculator with steps

Type a function of x and the point it approaches, a number, pi or infinity, and get the limit with the method explained: direct substitution, factoring and cancelling, L’Hôpital’s rule, or leading terms at infinity. Each answer is checked against a careful numerical approach from both sides.

Result

1 (decimal)

Steps

Substituting x = 0 gives 0/0, an indeterminate form
L'Hôpital's rule: differentiate the top and bottom separately, giving cos(x) / 1
Substitute x = 0: 1 / 1 = 1

Graph the function near the point

How to use it

1
Type the function: sin(x)/x, (x^2 - 1)/(x - 1), (3x^2 + 1)/(2x^2 - x).
2
Enter the point x approaches: a number like 1, a constant like pi/2, or inf and -inf for infinity.
3
Press Find the limit. If the two one-sided limits disagree, the calculator tells you the limit does not exist and gives both.

The methods, and when each one applies

Situation	Method	Example
Continuous at the point	Substitute directly	lim x→2 of x² + 1 = 5
0/0 with polynomials	Factor and cancel	lim x→1 of (x² − 1)/(x − 1) = 2
0/0 or ∞/∞ otherwise	L’Hôpital’s rule	lim x→0 of sin(x)/x = 1
Rational function at ∞	Divide by the highest power	lim x→∞ of (3x² + 1)/(2x² − x) = 3/2
Nonzero/0	Check the sign on each side	lim x→0 of 1/x² = ∞

Indeterminate forms

Substituting into (x² − 1)/(x − 1) at x = 1 gives 0/0. That doesn’t mean the limit is 0, 1 or undefined: it means substitution alone can’t decide. Both the top and bottom are heading to zero, and what matters is how fast each one gets there.

Factoring answers the question here: x² − 1 = (x − 1)(x + 1), and the (x − 1) cancels. The cancelled function, x + 1, equals the original everywhere except x = 1 itself, and a limit never looks at the point itself, only near it. So the limit is 1 + 1 = 2.

The other indeterminate forms, ∞/∞, 0·∞, ∞ − ∞, 1^∞, 0⁰ and ∞⁰, are usually rewritten as a quotient first and then handled the same way.

L’Hôpital’s rule

If f(x)/g(x) gives 0/0 or ∞/∞ at a point, then the limit of f/g equals the limit of f′/g′, as long as that second limit exists. You differentiate the top and the bottom separately (this is not the quotient rule).

For sin(x)/x at 0: differentiating gives cos(x)/1, and cos(0) = 1. For (1 − cos x)/x² at 0, one application gives sin(x)/(2x), still 0/0, and a second gives cos(x)/2, which is 1/2.

The rule needs the 0/0 or ∞/∞ form to start with. Applying it to anything else gives wrong answers, which is why the calculator always tests the form first.

Limits at infinity

For a rational function, divide the top and bottom by the highest power of x in the denominator. Every term with x left in a denominator goes to 0, and what survives decides the limit:

If the top has the lower degree, the limit is 0.
If the degrees are equal, it is the ratio of the leading coefficients.
If the top has the higher degree, the function grows without bound.

The same comparison of growth rates explains ln(x)/x → 0: by L’Hôpital the quotient becomes (1/x)/1, so a logarithm grows more slowly than any power of x.

How the answer is checked

Independently of the algebra, the calculator approaches the point from both sides with step sizes halving towards zero, and uses Richardson extrapolation to estimate where the values are heading. The steps are shown only when the algebra and this numerical limit agree. If the left and right limits differ, the limit does not exist, and both one-sided values are reported.

Examples to try

sin(x)/x near 0

y = sin(x)/x

Undefined at 0, but the graph heads to 1 from both sides.

Open this example

A removable hole

y = (x^2 - 1)/(x - 1)

The line y = x + 1 with a single missing point at x = 1. The limit is 2.

Open this example

A horizontal asymptote

y = (3x^2 + 1)/(2x^2 - x)y = 1.5

The limit at ±∞ is 3/2, the ratio of the leading coefficients.

Open this example

Frequently asked questions

Does the limit calculator show steps?

Yes. It shows which method applies (direct substitution, factor and cancel, L’Hôpital’s rule, or dividing by the highest power at infinity) and each step of it.

How do I enter infinity?

Type inf for +∞ and -inf for −∞ in the point box.

What if the limit doesn’t exist?

The calculator approaches from both sides. If the one-sided limits differ, it says the limit does not exist and gives both values. If the function grows without bound, the answer is ∞ or −∞.

When can I use L’Hôpital’s rule?

Only when substituting gives 0/0 or ∞/∞. Then the limit of f/g equals the limit of f′/g′, if that exists. The calculator checks the form before applying the rule.

Why is lim sin(x)/x = 1?

Near 0, sin(x) is almost exactly x (its Taylor series is x − x³/6 + …), so the ratio approaches 1. L’Hôpital’s rule gives the same: cos(0)/1 = 1.