Comparison of Infinitesimals Calculator | Equivalent Infinitesimals

lim α/β = c≠0 (same order); =1 (equivalent); =0 (higher order); =∞ (lower order)

Infinitesimal α(x)

Tends to 0 at a

Infinitesimal β(x)

Tends to 0 at a (denominator)

Point a

Both tend to 0 here

📖 Tutorial｜Comparison of Infinitesimals

1. Definition

Let α,β be infinitesimals in the same process. lim α/β=0 means α is higher order; =∞ lower; =c≠0 same order; =1 equivalent (α~β).

2. Symbols

Symbol	Meaning
α, β	Two infinitesimals
α/β	Their ratio
c	Nonzero limit of the ratio
a	Limit point

3. How it works

We compute α/β at a+h with shrinking h;
Watch whether the ratio tends to 0, a constant or 1;
Then classify the order relation and equivalence.

4. Steps

Enter α(x), β(x), both tending to 0 at a;
Enter a (usually 0);
Click Calculate to read the ratio trend.

5. Example

Example: as x→0, sinx/x→1 so sinx~x; x²/x=x→0 so x² is higher order than x.

6. Pitfalls

Equivalent infinitesimal replacement works for multiplicative factors, not arbitrary additive terms;
β must tend to 0 but stay nonzero (denominator);
Order is compared only within the same limit process.

❓ FAQ｜Comparison of Infinitesimals

Can I freely replace equivalent infinitesimals?

Only multiplicative factors. Replacing inside sums/differences often fails, e.g. (tanx−sinx)/x³.

What are common equivalent infinitesimals?

As x→0: sinx~tanx~arcsinx~arctanx~eˣ−1~ln(1+x)~x; 1−cosx~x²/2.

Same order vs equivalent?

Same order: ratio→c≠0. Equivalent is the special case c=1.

Why use very small h?

To observe the limit of α/β near a; smaller h is closer to the true limit.

What does “higher order” mean?

α shrinks to 0 much faster than β, e.g. x² vs x.