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Equivalent infinitesimal

When two variables α {\displaystyle \alpha } and β {\displaystyle \beta } converge to zero at the same limit point and lim β α = 1 {\displaystyle \textstyle \lim {\frac {\beta }{\alpha }}=1}, they are called equivalent infinitesimal (equiv. α ∼ β {\displaystyle \alpha \sim \beta }).

Moreover, if variables α ′ {\displaystyle \alpha '} and β ′ {\displaystyle \beta '} are such that α ∼ α ′ {\displaystyle \alpha \sim \alpha '} and β ∼ β ′ {\displaystyle \beta \sim \beta '}, then:

Here is a brief proof:

Suppose there are two equivalent infinitesimals α ∼ α ′ {\displaystyle \alpha \sim \alpha '} and β ∼ β ′ {\displaystyle \beta \sim \beta '}.

For the evaluation of the indeterminate form 0 / 0 {\displaystyle 0/0}, one can make use of the following facts about equivalent infinitesimals (e.g., x ∼ sin x {\displaystyle x\sim \sin x} if x becomes closer to zero):

For example:

In the 2nd equality, e y − 1 ∼ y {\displaystyle e^{y}-1\sim y} where y = x ln 2 + cos x 3 {\displaystyle y=x\ln {2+\cos x \over 3}} as y become closer to 0 is used, and y ∼ ln (1 + y) {\displaystyle y\sim \ln {(1+y)}} where y = cos x − 1 3 {\displaystyle y={{\cos x-1} \over 3}} is used in the 4th equality, and 1 − cos x ∼ x 2 2 {\displaystyle 1-\cos x\sim {x^{2} \over 2}} is used in the 5th equality.