evaluate limit by l'hospitals

Let y = (1 + a/x)bx, then


圖片參考：http://i388.photobucket.com/albums/oo325/loyitak1990/Nov09/Crazylim2.jpg

Hence, lim (x→∞) y = eab

Let y=(1+a/x)^(bx), take ln, lny=bx(1+a/x)

lim (x->∞) lny
=lim (x->∞) (1+a/x)/(1/bx) [0/0 type]
=lim (x->∞) [(-a/x^2)/(1+a/x)]/[-1/(bx^2)]
=lim (x->∞) [ab/(1+a/x)]
=ab

So lim (x->∞) (1+a/x)^(b x) =e^[ab]