一道高数微分题目,用泰勒公式做Lim(n->0)(x^2/2+1-(1+x^2)^(1/2))/(x^2*(sinx)^2)用泰勒公式求解

Question

一道高数微分题目,用泰勒公式做
Lim(n->0)(x^2/2+1-(1+x^2)^(1/2))/(x^2*(sinx)^2)用泰勒公式求解

Answer 1

分母 (x^2*(sinx)^2) x^4
(1+x^2)^(1/2)) = 1 + (1/2) x^2 + (1/2)(3/2) x^4 / 2!+ o(x^4)
= 1 + (1/2) x^2 + (3/8) x^4 + o(x^4)
分子 x^2/2+1 - (1+x^2)^(1/2) = - (3/8) x^4 + o(x^4)
原式 = lim(x->0) [- (3/8) x^4 + o(x^4) ] / x^4
= -3/8