Question

求下列函数的极限。

Answer 1

(2)

y->0

siny = y -(1/6)y^3 +o(y^3)

y-siny =(1/6)y^3 +o(y^3)

lim(x->∞) x^2.[ 1- xsin(1/x) ]

y=1/x

=lim(y->0) [ 1- siny/y ]/y^2

=lim(y->0) [ y- siny ]/y^3

=lim(y->0) (1/6)y^3/y^3

=1/6

(3)

x->0

sinx = x -(1/6)x^3 +o(x^3)

x-sinx =(1/6)x^3 +o(x^3)

lim(x->0) [1/sinx - 1/x ]

=lim(x->0) (x -sinx)/(xsinx)

=lim(x->0) (1/6)x^3/x^2

=0

(4)

lim(x->0) [1/x^2 - (cscx)^2 ]

=lim(x->0)  [(sinx)^2 - x^2 ]/[x^2 .(sinx)^2] ]

=lim(x->0)  [(sinx)^2 - x^2 ]/x^4                        (0/0分子分母分别求导）

=lim(x->0)  [sin(2x) - 2x ]/(4x^3)                        (0/0分子分母分别求导）

=lim(x->0)  [2cos(2x) - 2 ]/(12x^2)   

cos2x = 1- (1/2)(2x)^2 +o(x^2)

=lim(x->0)  {  2[ ( 1- (1/2)(2x)^2] - 2  }/(12x^2)   

=lim(x->0)    -4x^2/(12x^2)   

=-1/3

追问

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Answer 2

y, "Shall we count that miney one more time?" He grew exc