Lecture 11: Comparing Orders of Infinitesimals

11.1 Infinitesimal Comparative Example Problems

Example: As $x\to0$, let $\alpha(x)=kx^2$ and $\beta(x)=\sqrt{1+x\arcsin x}-\sqrt{\cos x}$. If they are equivalent infinitesimals, find $k$.

• Analysis
• Since it is an equivalent infinitesimal, the ratio is 1;

-When dealing with radicals differences, you can consider rationalizing and equivalent substitution methods for numerators;

• $=\frac1k\lim_{x\to0}\frac{1+x\arcsin x-\cos x}{x^2[\sqrt{1+x\arcsin x}+\sqrt{\cos x}]}$ (Physics and Chemistry)
• $=\frac1{2k}\lim_{x\to0}\frac{1+x\arcsin x-\cos x}{x^2}$
• Then split and present;
• Analysis
• Question Type: infinitelysmall

Two Equivalent Infinitesimals in a More General Form

• $\log^{a}(1+2x)\sim(2x)^{a}=2^{a}x^{a}$
• $(1-ax)^{\frac{1}{\alpha}}\sim(\frac{1}{2}x^{2})^{\frac{1}{\alpha}}$

Infinitesimal Order Sorting Problem

• Method 1: Pair up and compare;
• Method 2: Set the order for each one;
• For example: $\alpha_1=x(\cos\sqrt{x}-1),\alpha_2=\sqrt{x}\ln(1+\sqrt[3]{x}),\alpha_3=\sqrt[3]{x+1}-1.$ When $x\to0^+$, the order of the above three infinitesimals from lower to higher order is:
• You can assign the order to a1, a2, a3, and then sort;