題目 Problem Find the following limits. Show your work. (a) (5 pts) limx→∞sin(x)x\displaystyle \lim_{x\to\infty}\frac{\sin(x)}{x}x→∞limxsin(x). (b) (5 pts) limh→0(2+h)cos(2+h)−2cos(2)h\displaystyle \lim_{h\to 0}\frac{(2+h)\cos(2+h)-2\cos(2)}{h}h→0limh(2+h)cos(2+h)−2cos(2). (c) (5 pts) limx→∞∫1xln(t) dt∫1xtln(t) dt\displaystyle \lim_{x\to\infty}\frac{\int_1^x \ln(t)\,dt}{\int_1^x t\ln(t)\,dt}x→∞lim∫1xtln(t)dt∫1xln(t)dt. (d) (5 pts) limn→∞1n∑k=1nln (1+2kn)\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n \ln\!\left(1+\frac{2k}{n}\right)n→∞limn1k=1∑nln(1+n2k). 解答 待補。
