Advanced Math

Limit Calculator

Finds exact limits of polynomial and rational expressions by direct substitution, by factoring and cancelling a common (x − a) when substitution gives 0/0, and by comparing leading terms at ±∞. One-sided limits and vertical asymptotes are handled. It also knows the standard limit sin(kx)/(mx) → k/m as x → 0.

Inputslim x→a f(x)

f(x)Polynomials and fractions of polynomials, e.g. (x^2 − 1)/(x − 1), (3x^2 + 1)/(x^2 − 5), or sin(3x)/(2x).x approachesA number, a fraction, inf or −inf.Direction

Step-by-step

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Fill in the fields and press Solve. The full method will appear here, one numbered step at a time.

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Worked example

Find the limit of (x² − 4)/(x − 2) as x → 2.

1.
Write down the limit.
```
lim(x → 2) (x^2 − 4)/(x − 2)
```
2.
Write it as one fraction of polynomials.
```
(x^2 − 4)/(x − 2)
```
3.
Substitute x = 2.
```
numerator = 0,  denominator = 0
```
4.
Both are 0, giving the indeterminate form 0/0. By the factor theorem, (x − 2) is a factor of both. Factor it out.
```
numerator = (x − 2)(x + 2),  denominator = (x − 2)
```
5.
Cancel (x − 2). This is allowed because a limit only looks at x close to 2, never at x = 2 itself.
```
x + 2
```
6.
Substitute x = 2.
```
4
```
7.
The denominator is no longer 0, so substitute directly.
```
= 4
```

Answer: lim(x → 2) (x^2 − 4)/(x − 2) = 4

Note: Exact working for polynomials and fractions of polynomials, plus sin(kx)/(mx) and tan(kx)/(mx) at 0.

Use “Load worked example” above to run this in the solver.