lim(cosx-cos3x)/(5x) x趋向0

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lim(cosx-cos3x)/(5x)x趋向0lim(cosx-cos3x)/(5x)x趋向0lim(cosx-cos3x)/(5x)x趋向0原式=lim(x->0)[(cosx-cos(3x))''

lim(cosx-cos3x)/(5x) x趋向0
lim(cosx-cos3x)/(5x) x趋向0

lim(cosx-cos3x)/(5x) x趋向0
原式=lim(x->0)[(cosx-cos(3x))'/(5x)'] (0/0型极限,应用罗比达法则)
=lim(x->0)[(3sin(3x)-sinx)/5]
=(3sin0-sin0)/5
=0.

因为:
cosx-cos3x
=cosx-4(cosx)^3+3cosx
=4cosx-4(1-(sinx)^2)cosx)
=4cosx-4cosx+4cosx(sinx)^2
=4cosx(sinx)^2
当x趋近0时 limx=limsinx
所以 lim(cosx-cos3x)/5x=lim4cosx(sinx)^2/5sinx=4sinx/5=0

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