Limit Calculator

Limit calculator that tells you which case you are in

Enter a function and the value x is approaching. You get the limit, the reason that method applies, and a clear answer when the limit is infinite or does not exist, rather than a number with no explanation attached.

Part of SolveStep's calculus solver. Handles limits at a point, at infinity, and from one side.

Direct substitution, 0/0 by cancelling, limits at infinity, and one-sided limits

What a limit actually asks

The value the function is heading for, not the value it has

A limit asks a question about the neighbourhood of a point, never about the point itself. (x² − 4)/(x − 2) has no value at all when x = 2, because that would be 0/0, and yet the limit there is a perfectly ordinary 4. Those two facts are not in conflict. The function is undefined at 2 and heading straight for 4 on both sides of it.

That distinction is why cancelling a factor is legal inside a limit when it would not be legal as an identity. Dividing top and bottom by (x − 2) changes the function at exactly one point, and that one point is the one the limit ignores.

Three outcomes are possible, and this calculator names which one you have rather than printing a number and leaving you to guess: the limit is a finite value, the limit is infinite, or the limit does not exist because the two sides disagree.

Substitution works

lim(x→1) (2x+1)/(x²+1) = 3/2

The denominator is not zero at the point, so the function is continuous there and you simply put the number in.

0/0, so factor and cancel

lim(x→2) (x²−4)/(x−2) = 4

Both parts vanish, which guarantees a shared factor. Cancel it, then substitute into what is left.

Bottom zero, top not

lim(x→0) 1/x does not exist

It runs to −∞ from the left and +∞ from the right. A two-sided limit needs both sides to agree.

x heading to infinity

lim(x→∞) (3x²+2x)/(x²−1) = 3

Compare degrees. Equal degrees give the ratio of the leading coefficients and nothing else matters.

Method

How to evaluate a limit, in the order you should try things

Substitute first.

It is right far more often than students expect, and it takes two seconds. If the denominator is nonzero at the point, the function is continuous there and the substituted value is the limit.

If you get 0/0, factor.

Zero on top and zero on the bottom at x = a guarantees that (x − a) divides both. Cancel it and substitute again. This one technique handles the large majority of limit questions set before L'Hôpital's rule appears.

If you get a nonzero number over 0, check each side separately.

The magnitude runs to infinity; the only question left is the sign, and the sign can differ left and right. When it does, the limit does not exist.

If x is going to infinity, compare degrees.

Bottom bigger, the limit is 0. Degrees equal, the limit is the ratio of the leading coefficients. Top bigger, it is infinite, and the sign follows the leading coefficients and the parity of the degree gap.

If it is one of the standard forms, use it.

sin(x)/x → 1 and (1 − cos x)/x → 0 as x → 0 are meant to be recognised, not re-derived. They are the limits that make the trig derivatives work.

Worked examples

Three limits, worked the way you would write them

The 0/0 case

lim(x→3) (x² − 9)/(x² − x − 6)

Substitute: the top is 9 − 9 = 0 and the bottom is 9 − 3 − 6 = 0, so this is the 0/0 case.
Both are divisible by (x − 3). The top factors as (x − 3)(x + 3), the bottom as (x − 3)(x + 2).
Cancel: the expression becomes (x + 3)/(x + 2) everywhere except at x = 3 itself.
Substitute into the simplified form: 6/5.

6/5

A limit at infinity

lim(x→∞) (2x + 1)/(x² + 5)

Degree 1 on top, degree 2 on the bottom.
Divide every term by x²: the expression becomes (2/x + 1/x²) / (1 + 5/x²).
As x grows, every term with x underneath goes to 0, leaving 0/1.
The bottom outgrows the top, which is the general rule whenever the denominator has the higher degree.

When the limit does not exist

lim(x→2) 1/(x − 2)

Substitute: the bottom is 0, the top is 1, so the value is unbounded.
Approach from the left, at x = 1.999: the bottom is a small negative number, so the quotient is a large negative number, heading to −∞.
Approach from the right, at x = 2.001: the bottom is a small positive number, so the quotient heads to +∞.
The two sides disagree, so the two-sided limit does not exist. There is a vertical asymptote at x = 2. The one-sided limits do exist and are worth stating separately.

does not exist (−∞ from the left, +∞ from the right)

Try your own limit above →

Where students lose marks

Three mistakes that account for most wrong answers

Writing "∞" when the answer is "does not exist". These are different claims. 1/x² at 0 really does run to +∞ from both sides, so ∞ is the correct description. 1/x at 0 does not, because the sides disagree, and writing ∞ there is marked wrong.

Treating 0/0 as if it were 0, or as if it were 1. It is neither. The expression 0/0 is called indeterminate precisely because the answer depends entirely on which functions produced the zeros. (x²−4)/(x−2) and (x−2)/(x²−4) both give 0/0 at x = 2 and their limits are 4 and 1/4.

Substituting into the simplified form and forgetting it was simplified. The limit is 4, but the function is still undefined at x = 2. If the question asks for the value of the function rather than the limit, the answer is "undefined".

Scope

What this calculator will and will not do

It handles polynomials and rational functions exactly, at a finite point or at ±∞, including one-sided limits and the 0/0 case with repeated factors. It also recognises the standard limits sin(x)/x, tan(x)/x, (1 − cos x)/x and (eˣ − 1)/x at 0.

It does not yet handle general trig, logarithmic or exponential limits, limits of piecewise functions, or sequences. When it cannot do one, it says so by name instead of producing a number that looks plausible. A limit calculator that guesses is worse than useless, because you have no way to tell a guess from a result.

For the derivative that a limit defines, use the derivative calculator. For the vertical and horizontal asymptotes that infinite limits describe, the asymptote calculator finds all of them at once.

Common questions

Limit calculator FAQ

What is a limit in calculus?

A limit describes the value a function approaches as its input approaches some number, whether or not the function is defined at that number. It is written lim(x→a) f(x). The distinction matters: (x² − 4)/(x − 2) is undefined at x = 2 and its limit there is 4.

How do you find a limit by hand?

Substitute the value first. If the denominator is not zero, that answer is the limit. If you get 0/0, factor the top and bottom, cancel the shared (x − a) factor, and substitute into what remains. If you get a nonzero number over zero, the limit is infinite and you check each side for the sign.

What does it mean when a limit does not exist?

It means the function approaches different values from the two sides, so no single value describes the behaviour. lim(x→0) 1/x does not exist because the function runs to −∞ from the left and +∞ from the right. Both one-sided limits exist; the two-sided one does not.

Is an infinite limit the same as a limit that does not exist?

Strictly, an infinite limit is a kind of non-existence, because ∞ is not a number. In practice most courses want you to write ∞ when the function grows without bound in the same direction from both sides, and "does not exist" when the two sides disagree. This calculator distinguishes the two cases.

How do you find a limit at infinity?

Compare the degrees of the numerator and denominator. If the bottom degree is larger, the limit is 0. If the degrees are equal, the limit is the ratio of the leading coefficients. If the top degree is larger, the limit is +∞ or −∞ depending on the signs and the parity of the degree gap.

Why can you cancel a factor that equals zero at the point?

Because a limit only looks at values near the point, never at the point itself. Dividing by (x − a) changes the function at exactly one input, and the limit ignores that input by definition. This is why cancelling is valid inside a limit even though it is not a valid simplification of the function.

Does this limit calculator show the steps?

Yes, every one. It names which case applies, shows the cancelled factor or the degree comparison, and explains why the step is allowed. There is no paid tier and no account.

Can it do L'Hôpital's rule?

It does not apply L'Hôpital's rule, because for rational functions the factoring method reaches the same answer and shows working that a first-year course will accept. For 0/0 cases involving logs, exponentials or general trig, work the derivative of the top and bottom yourself with the derivative calculator and evaluate the new quotient.

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