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lim(tAnx%sinx/sinx*3)

lim(x→0)(tanx-sinx)/(sinx)^3 =lim(x→0)tanx(1-cosx)/x^3 =lim(x→0)x*(1/2x^2)/x^3 =1/2

lim(tanx-sinx/sin³x)=lim(1-cosx)/sin²x=lim 2sin²(x/2)/sin²x=(x²/2)/x²=1/2 x→0

1/2 分子tanx-sinx=sinx(1-cos)/cosx~x·x^2/2=x^3/2; 分母sinx^3~x^3; 所以极限为1/2

见图。

LIM(X趋向0)tanx-sinx/x3次方=lim(x->0)tanx(1-cosx)/x³=lim(x->0)(x·x²/2)/x³=1/2

因为tanx精确的说是近似与x/√(x*x+1),这类应用无穷近似值求极限不能简单粗暴的把lim(A+B)拆成=limA+limB,而得先应用三角函数转化,把(tanx-sinx)=(1-cosx)sinx/cosx=2sin(x/2)sin(x/2)sinx/cosx再应用替换,基本都必须把和差形变...

求极限x➔0lim[(tanx-sinx)/sin³x] =lim(1/cosx-1)/(sinx)^2 =lim(1-cosx)/(sinx)^2cosx =lim2(sin(x/2))^2/(sinx)^2 =(1/2)lim[(sin(x/2))^2/(x/2)^2][x^2/(sinx)^2] =1/2

猜x→0时 (tanx-sinx)/x^3 →(sec^x-cosx)/(3x^2) →[(-2)(-sinx)/(cosx)^3+sinx]/(6x) →1/2.

无穷近似值代换,二倍角公式 =lim(1-cosx)/x²limtanx/x =lim2sin²(x/2)/x² =lim2(x/2)²/x² =1/2

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