高数:一道极限题 cos(sinx)
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高数:一道极限题 cos(sinx)
高数:一道极限题 cos(sinx)
高数:一道极限题 cos(sinx)
lim[cos(sinx)-cosx]/x^4
cosx=1-(1/2)x^2+(1/24)x^4+o(x^4)
sinx=x-(1/6)x^3+o(x^3)
cos(sinx)=1-(1/2)(x-(1/6)x^3)^2+(1/24)x^4+o(x^4)
=1-(1/2)x^2+(1/6)x^4+(1/24)x^4+o(x^4)
原式=lim[1-(1/2)x^2+(1/6)x^4+(1/24)x^4-(1-(1/2)x^2+(1/24)x^4)+o(x^4)]/x^4
=1/6
cosx=1-1/2 x² +1/24 x^4 +o(x^4)
sinx=x-1/6 x³ +o(x^4)=x+o(x)
∴cos(sinx)=1-1/2 (x-1/6 x³ +o(x^4))² +1/24 (x+o(x))^4+o(x^4)=1-1/2 x² +5/24 x^4
∴cos(sinx)-cosx=1/6 x^4+o(x^4)
(sinx)^4=(x+o(x))^4=x^4+o(x^4)
∴lim(cosxinx-cosx)/(sinx)^4=1/6