Question

求极限，要步骤

limn^2{arctan(1/n)-arctan(1/1+n)}
n→无穷。

Answer 1

方法很多，这里用中值定理来做
由中值定理有存在c,1/n<c<1/(1+n),使arctan(1/n)-arctan(1/(1+n))= 1/(1+c²)(1/n -1/(1+n) )
显然有n→∞时，c→0
所以原式 = lim 1/(1+c²)n²(1/n -1/(1+n) )
=lim 1/(1+c²)(n/(1+n) )
=lim 1/(1+c²) * lim (n/(1+n) )
=1*1=1