The first higher number: infinitesimal comparison Higher Education Press: Advanced Mathematics (Volume I) P59 original sentence: the different results of the quotient of two infinitesimals reflect the differences in the "speed" of different infinitesimals tending to zero. Q: for example, if x / X indicates that x is of higher order, then x tends to zero faster than x, but when x belongs to 0 to 1 / 2, X falls faster than x under the same X difference,

The first higher number: infinitesimal comparison Higher Education Press: Advanced Mathematics (Volume I) P59 original sentence: the different results of the quotient of two infinitesimals reflect the differences in the "speed" of different infinitesimals tending to zero. Q: for example, if x / X indicates that x is of higher order, then x tends to zero faster than x, but when x belongs to 0 to 1 / 2, X falls faster than x under the same X difference,

x 1 1/2 1/3 1/4...
y1=x 1 1/2 1/3 1/4
y2=x^2 1 1/4 1/9 1/16 ...
When X -- > 0, x ^ 2 is the higher order infinitesimal of X, even if x ^ 2 tends to 0 faster, it should be compared that the independent variable x decreases by the same amount, which is closer to the distance between X ^ 2 and X from 0, and you really see the difference between the two dependent variables, namely △ Y1, △ Y2

Using the property of infinitesimal, we can find the limit limx → 0 〔√ (1 + xtanx)] - 1 / (1-cosx)