Limit Laws Calculator | Sum Difference Product Quotient

lim[f±g]=lim f±lim g lim[fg]=lim f·lim g lim[f/g]=lim f/lim g

Function f(x)

Supports + - * / ^ and sin cos tan sqrt exp log pi e

Function g(x)

Same as above

Point a

x approaches this point

📖 Tutorial｜Limit Laws

1. Definition

If lim f(x)=A and lim g(x)=B both exist, the limit of a sum, difference, product, quotient, constant multiple or power equals the corresponding operation on A and B.

2. Symbols

Symbol	Meaning
f(x), g(x)	The two functions
a	The point x approaches
A, B	Limits of f and g

3. How it works

We numerically approximate A and B via f(a±h);
Then apply arithmetic and powers to A and B to illustrate the laws;
The quotient law requires B≠0.

4. Steps

Enter f(x), e.g. x^2;
Enter g(x), e.g. 2*x;
Enter the point a, e.g. 3;
Click Calculate to see each law.

5. Example

Example: lim(x²+2x) as x→3.
Solution: lim x²=9, lim 2x=6, so lim(x²+2x)=15. The tool shows sum=15, product=54, quotient=1.5.

6. Pitfalls

The arithmetic laws require both limits to exist and be finite;
If the denominator limit is 0, do not divide directly—factor or use L’Hôpital;
Numerical approach has small rounding error.

❓ FAQ｜Limit Laws

When can I not use the quotient law?

When lim g(x)=0, the denominator vanishes and the law fails. Factor out the common zero factor, or use L’Hôpital or equivalent infinitesimals.

If both limits exist, is the limit of a difference the difference of limits?

Yes. As long as lim f and lim g both exist finitely, lim(f−g)=lim f−lim g.

Why is my result slightly off?

The numerical approach uses f(a±h) with the last h=0.0001, giving floating-point noise that does not change the conclusion.

How does the constant multiple law work?

lim[c·f(x)] = c·lim f(x); the constant c can be pulled outside the limit.

Is the power law only for squares?

No. lim[f(x)]ⁿ = (lim f(x))ⁿ holds for positive integers n, and more generally.