求 $\dps{\lim_{t\to +\infty}\sex{\frac{1}{t} +\frac{2t}{t^2+1^2}+\frac{2t^2}{t^2+2^2}+\cdots+\frac{2t}{t^2+n^2}+\cdots}}$.
解答: $$\beex \bea \mbox{原极限}&=\vlm{t}\vsm{n} \frac{2t}{t^2+n^2}\quad\sex{\vlm{t}\frac{1}{t}=0}\\ &=\lim_{h\to 0^+} \vsm{n}\frac{2h}{1+(nh)^2}\quad\sex{t=\frac{1}{h}}\\ &=2\int_0^\infty \frac{1}{1+x^2}\rd x\quad\sex{\mbox{例 5.1.55}}\\ &=\pi. \eea \eeex$$
