Question

高等数学求极限，题目如图，求个过程，谢谢啦？

Answer 1

(1)
lim(x->0) sinx/x =1
(2)
lim(x->∞) sinx/x =0
(3)
lim(x->∞) xsin(1/x)
y=1/x
=lim(y->0+0) siny/y
=1
(4) lim(x->0) x.sin(1/x) =0

追问

我只是想，要个过程😂

追答

(1)
lim(x->0) sinx/x =lim(x->0) x/x =1
(2)
|sinx|≤1
-1/x ≤sinx/x≤ 1/x
lim(x->∞) -1/x = lim(x->∞) 1/x =0
=> lim(x->∞) sinx/x =0
(3)
lim(x->∞) xsin(1/x)
y=1/x
=lim(y->0+) siny/y
=lim(y->0+) y/y
=1
(4)
|sin(1/x)|≤1
-x ≤xsin(1/x)≤ x
lim(x->0) -x = lim(x->0) x =0
=> lim(x->0) x.sin(1/x) =0

追问

谢谢您嘞