一道求极限limx->a(sinx/sina)^(1/(x-a))

Question

一道求极限
lim x->a (sinx/sina)^(1/(x-a))

Answer 1

lim(x→a) (sinx/sina)^[1/(x-a)]
=lim(x→a) [1+(sinx/sina-1)]^[1/(x-a)]
=lim(x→a) [1+(sinx/sina-1)]^{[1/(sinx/sina-1)]*[(sinx/sina-1)/(x-a)]}
=e^lim(x→a) (sinx/sina-1)/(x-a)
=e^lim(x→a) (sinx-sina)/[(x-a)sina]
=e^lim(x→a) 2cos[(x+a)/2]sin[(x-a)/2]/[(x-a)sina]
=e^lim(x→a) 2cos[(x+a)/2][(x-a)/2]/[(x-a)sina]
=e^lim(x→a) cos[(x+a)/2]/sina
=e^(cota)