Infinitesimals

Definition

It is a variable that is infinitely approaching 0.

Common Equivalent Infinitesimals

These are equivalent infinitesimals that will appear in many cases. They do help a lot when defining limits and finding derivatives.

• ${\displaystyle \sin x\sim x}$
• ${\displaystyle \tan x\sim x}$
• ${\displaystyle \arcsin x\sim x}$
• ${\displaystyle \arctan x\sim x}$
• ${\displaystyle \ln (x+1)\sim x}$
• ${\displaystyle e^x-1\sim x}$
• ${\displaystyle x-\sin x\sim -\frac{1}{6}{x}^3}$
• ${\displaystyle 1-\cos x\sim \frac{1}{2}{x}^2}$
• ${\displaystyle a^x-1\sim x\ln a}$

Higher-Order Infinitesimals

It implies which infinitesimal is approaching 0 faster when two infinitesimals appears at the same time.

Determination

To define the higher-order or the equivalent infinitesimal, the general method is to find the limit of the ratio of two infinitesimals. Let ${\displaystyle \underset{x\to x_0}{lim}\frac{\alpha (x)}{\beta (x)}=L}$

• When ${\displaystyle L=0}$, ${\displaystyle \alpha }$ is the higher-order infinitesimal of ${\displaystyle \beta }$, denoted by ${\displaystyle \alpha =o(\beta )}$, which implies that ${\displaystyle \alpha }$ is approaching 0 faster then ${\displaystyle \beta }$.

• When ${\displaystyle 0<|L|<+\mathrm{\infty }}$, it shows that ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are in the same order, with a difference of a constant coefficient. We say they are equivalent infinitesimals.

• When ${\displaystyle L=\mathrm{\infty }}$, it shows that ${\displaystyle \underset{x\to x_0}{lim}\frac{\beta (x)}{\alpha (x)}=0}$, which tells us ${\displaystyle \beta =o(\alpha )}$.