Question

我是人類

想問這兩題怎麼解

sin(sin x)
sin x
sin(sin x)
(2) lim
x→0
(3) lim
x→0
X

Answer 1

qn

(2) 設 t=sinx
當 x→0 時，t→0
lim_{x→0} sin(sinx)/sinx
= lim_{t→0} sint/t
= 1

(3)
lim_{x→0} sin(sinx)/x
= lim_{x→0} [sin(sinx)·sinx]/[x·sinx]
= lim_{x→0} [sin(sinx)/sinx]·[sinx/x]
= [lim_{x→0} sin(sinx)/sinx]·[lim_{x→0} sinx/x]
= 1·1
= 1
(快一點的看法是：裡面的sinx近似於x，就能直接求出來)