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x趋近与0,tAn3x%Cosx+1/sin2x的极限,求个详细过程

在x趋于0的时候, sinx和tanx都等价于x 而1-cosx等价于0.5x^2 所以这里的 lim(x趋于0) (tan3x-cosx+1)/ sin2x =lim(x趋于0) tan3x/sin2x +(1-cosx)/sin2x =lim(x趋于0) 3/2 +0.5x^2 /2x =3/2

tanx的导数是(secx)^2,tan3x的导数是3(sec3x)^2洛比达法则要用两次原式=(1/3)*lim[(cos3x)/(cosx)]^2=(1/3)*lim[(-3sin3x)/(-sinx)]^2=3*lim{[sin(3π/2)/sin(π/2)]^2}=3

tanx=sin3x/cos3x=1/cot3x

∫ 1/(sin³xcosx) dx 分子分母同除以(cosx)^4 =∫ (secx)^4/(tan³x) dx =∫ sec²x/(tan³x) d(tanx) =∫ (tan²x+1)/(tan³x) d(tanx) =∫ 1/tanx d(tanx) + ∫ 1/(tan³x) d(tanx) =ln|tanx| - 1/(2tan²x) + C ...

当然算了

(sinx+sin3x)/(cosx+cos3x) =[2sin(x+3x)/2*cos(x-3x)/2]/(2cos(x+3x)/2*cos(x-3x)/2) =sin2xcosx/cos2xcosx =tan2x

'0/0'型分别对分子分母求倒!太简单了吧!

求极限 当x趋向于π/2时 limtanx/tan3x 解:lim(x→π/2)tanx/tan3x =lim(x→π/2)(sinx/cosx)/(sin3x/cos3x) =lim(x→π/2)(1/cosx)/((-1)/cos3x) =-lim(x→π/2)(cos3x/cosx) =-lim(x→π/2)(-3sin3x)/(-sinx) =3

tan3x=tan(2x+x) =(tan2x+tanx)/(1-tan2x·tanx) =【2tanx/(1-tanx的平方)+tanx】/【1-2tanx的平方/(1-tanx的平方)】 =(3tanx-tanx的3次方)/(1-3·tanx的平方)

tan3/2x-tanx/2=(cos(x/2)*sin(3x/2)-cos(3x/2)sin(x/2))/(cos(3x/2)*cos(x/2)) =sin(3x/2-x/2)/(1/2)(cos(3x/2+x/2)+cos(3x/2-x/2)) =2sinx/(cosx+cos2x)

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