Question

极限怎么做 66题求大佬指点

Answer 1

lim(n->∞) cos(x/2). cos(x/4)....cos(x/2^n)
分子分母同时乘以sin(x/2^n)
=lim(n->∞) cos(x/2). cos(x/4)....cos(x/2^n) . sin(x/2^n) /sin(x/2^n)
=lim(n->∞) cos(x/2). cos(x/4)....cos(x/2^(n-1)). [ (1/2)sin(x/2^(n-1) ) ] /sin(x/2^n)
=lim(n->∞) cos(x/2). cos(x/4)....cos(x/2^(n-2)). [ (1/2^2)sin(x/2^(n-2) ) ] /sin(x/2^n)
=...
=...
=lim(n->∞) (1/2^(n-1)) . cos(x/2).sin(x/2) /sin(x/2^n)
=lim(n->∞) (1/2^n) . sinx /sin(x/2^n)
=lim(n->∞) (1/2^n) . sinx /( x/2^n)
=sinx/x

Answer 2

求极限部分上下均大于0不等于1, 因此对求极限部分取自然对数后有：
[x^2*ln (1+1/x)] /x = x *[ln(1+x)-lnx] = [ln(1+x)-lnx] /(1/x)
上下同时求导有[1/(x+1) - 1/x]/ (-1/x^2) = x^2 *[1/x-1/(x+1)] = x/(x+1), 当x趋于+∞时，极限为1.
因此原极限为e.