求极限lim(1-cosxy)/x²y²,xy都趋于0

Question

求极限lim(1-cosxy)/x²y²,xy都趋于0

Answer 1

假设沿着 y = kx 趋近于 原点,则：
lim[1-cos(xy)]/(xy)^2
=lim[1-cos(kx)^2]/(k^2*x^4)
=lim 2{sin[(kx)^2/2]}^2/{[(kx)^2/2]^2 * 4/k^2}
=lim{sin[(kx)^2/2]}^2/{[(kx)^2/2]^2 * 2/k^2}
=lim{sin[(kx)^2/2]/[(kx)^2/2]}^2 /(2/k^2)
=lim 1/(2/k^2)
=k^2/2
因为 k 是任意直线的斜率,所以,上面的极限是不存在的.