请问1/(sinx)^2-(cosx)^2/x^2x趋近于0的极限为什么等于4/3?

Question

请问 1/(sinx)^2 -(cosx)^2/x^2 x趋近于0 的极限为什么等于4/3?

Answer 1

先通分,再因式分解与等价无穷小替换,最后用洛必达法则.
原式＝lim [x^2-(sinxcosx)^2)]/(xsinx)^2
＝lim (x+sinxcosx)(x-sinxcosx)/x^4
＝lim (x+sinxcosx)/x×(x-sinxcosx)/x^3
＝lim (1+sinx/x×cosx)/x × lim (x-sinxcosx)/x^3
＝2×lim (x-sinxcosx)/x^3
＝2×lim (1-cos2x)/(3x^2)
＝2×lim (2sin2x)/(6x)
＝2×2×2/6＝4/3