Question

用洛必达法则求下列极限

Answer 1

lim(x->∞) [(x^2+1)/(x^2-1)]^x^2
=lim(x->∞) [ 1 + 2/(x^2-1) ]^x^2
let
1/y=2/(x^2-1)
x^2-1 = 2y
x^2 = 2y+1

lim(x->∞) [(x^2+1)/(x^2-1)]^x^2
=lim(x->∞) [ 1 + 2/(x^2-1) ]^x^2
=lim(y->∞) [ 1 + 1/y ]^(2y+1)
=e^2

---------------------------------------------
x->0
(sinx)^3 ~ x^3
e^[(sinx)^3] ~ 1+ x^3
e^[(sinx)^3] - 1 ~ x^3

1-cosx ~ (1/2)x^2
x(1-cosx) ~(1/2)x^3

lim(x->0) [ e^[(sinx)^3] - 1 ]/[x(1-cosx) ]
=lim(x->0) x^3/[(1/2)x^3]
=2