lim{[3sinx+(x^2)*cos(1/x)]/[(1+cosx)ln(1+x)]}(x趋近于0)

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lim{[3sinx+(x^2)*cos(1/x)]/[(1+cosx)ln(1+x)]}(x趋近于0)lim{[3sinx+(x^2)*cos(1/x)]/[(1+cosx)ln(1+x)]}(x趋

lim{[3sinx+(x^2)*cos(1/x)]/[(1+cosx)ln(1+x)]}(x趋近于0)
lim{[3sinx+(x^2)*cos(1/x)]/[(1+cosx)ln(1+x)]}(x趋近于0)

lim{[3sinx+(x^2)*cos(1/x)]/[(1+cosx)ln(1+x)]}(x趋近于0)
原式=lim[3sinx+(x^2)cos(1/x)/2x]
=lim[3sinx/2x+xcos(1/x)/2]
=3/2+0=3/2
其中当x趋近于0时,1+cosx趋近于2;ln(1+x)和x等价无穷小;cos(1/x)为有界函数,所以xcos(1/x)/2=0;lim(sinx/x)=1

x→0,1+cosx→1,ln(1+x)和x等价无穷小
lim{[3sinx+(x^2)*cos(1/x)]/[(1+cosx)ln(1+x)]}
=lim [3sinx+(x^2)*cos(1/x)]/x
=lim 3sinx/x+x*cos(1/x)
=3+0=3

Matlab:
>> syms x
>> limit((3*sin(x)+(x^2)*cos(1/x))/((1+cos(x))*log(1+x)),x,0)

ans =

3/2