当x趋近于0时,\text{Ci}(x)的定义为:
[\text{Ci}(x) = -\int_x^\infty \frac{\cos t}{t} dt]
所以,要求解\lim_{x\to 0} [\text{Ci}(mx) - \text{Ci}(nx)],我们可以先求出\text{Ci}(x)的导数:
[\frac{d}{dx}[\text{Ci}(x)] = -\frac{\cos x}{x}]
然后使用洛必达法则来求解极限:
[\lim_{x\to 0} [\text{Ci}(mx) - \text{Ci}(nx)] = \lim_{x\to 0} \int_x^\infty \frac{\cos mt - \cos nt}{t} dt]
[= \int_0^\infty \lim_{x\to 0} \frac{\cos mx - \cos nx}{x} dt]
[= \int_0^\infty (n\sin nt - m\sin mt) dt]
[= \frac{n^2 - m^2}{n^2 + m^2}]
对于\text{Ci}(x) - \ln|x|,我们可以先求出\text{Ci}(x)和\ln|x|的导数:
[\frac{d}{dx}[\text{Ci}(x)] = -\frac{\cos x}{x}]
[\frac{d}{dx}[\ln|x|] = \frac{1}{x}]
然后计算\text{Ci}(x) - \ln|x|的导数:
[\frac{d}{dx}[\text{Ci}(x) - \ln|x|] = -\frac{\cos x}{x} - \frac{1}{x}]
最后,我们可以计算极限\lim_{x\to 0} [\text{Ci}(x) - \ln|x|],这个极限需要用到洛必达法则。