$\int_ϵ^1{\cos x\over x}\rmd x$的估计

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对于$0<ϵ<1$ 当$ϵ\to0$ 估计$f(ϵ)=\int_ϵ^1{\cos x\over x}\rmd x$
由$\cos x=1+O(x^2)$得$$\int_ϵ^1{\cos x\over x}\rmd x=\int_ϵ^1\frac1x+O(x)\rmd x=\log\frac1ϵ+O(ϵ^2)$$
上面我们使用了L'hospital推出 $\lim_{x\to0}\frac{f(x)}{g(x)}=1\implies\lim_{x\to0}\lim_{ε\to0}\frac{\int_ε^1f(x)\rmd x}{\int_ε^1g(x)\rmd x}=1$
验证:

Limit[Integrate[Cos[x]/x,{x,ϵ,1},Assumptions->0<ϵ<1]/Log[1/ϵ],ϵ->0]

输出1

画图:
import graph;
unitsize(8cm,1cm);

real f(real x) { return cos(x)/x; }
path g = graph(f, 0.1, 1);

xaxis("$x$",Ticks("%",Step=0.2,Size=.5mm),
      p=opacity(0.7));
yaxis("$y$",Ticks("%",Step=0.2,Size=.5mm),
      p=opacity(0.7));

fill(g--(1,0)--(0,0)--(0,point(g,0).y)--cycle, blue+opacity(0.3));
draw(g, blue+linewidth(1.2pt));