利用等价无穷小求极限:lim(x→0)(cosx+2sinx)^(1/x)

问题描述:

利用等价无穷小求极限:lim(x→0)(cosx+2sinx)^(1/x)

lim(x→0) (cosx+2sinx)^(1/x)
=lim(x→0) [1+(cosx-1+2sinx)]^(1/x)
=lim(x→0) {[ 1 + (cosx-1+2sinx) ]^[1/(cosx-1+2sinx)]}^[(cosx-1+2sinx)(1/x)]
∵lim(x→0) {[ 1 + (cosx-1+2sinx) ]^[1/(cosx-1+2sinx)]} = e
∵lim(x→0) [(cosx-1+2sinx)(1/x)]
=lim(x→0) [cosx-1]/x + lim(x→0) 2sinx/x
=lim(x→0) [-x^2/2]/x + lim(x→0) 2x/x
= 0+2 = 2
= e^2