Question

求(x-tanx)/(xsin^2x)的极限

lim(x趋于0)x^2e^(1/x^2) lim(x趋于0+)x^sinx

Answer 1

1、原式=limx→0 (x-tanx)/x^3 (sinx~x替换)
=limx→0 (1-1/cos^2x)/3x^2
=limx→0 -sin^2x/3x^2
=limx→0 -x^2/3x^2
=-1/3；
2、原式=limx→0 e^(1/x^2)/(1/x^2)
=limn→∞ e^n/n
=limn→∞ e^n/1
=无穷大；
3、原式=limx→0+ e^(sinx*lnx)
=e^limx→0+ lnx/(1/sinx)
=e^limx→0+ (1/x)/[(-1/sin^2x)*cosx]
=e^limx→0+ -sin^2x/(x*cosx)
=e^limx→0+ -x^2/x
=e^0
=1。