lim（x趋近于0）（cos2x-cos3x）/x²

Question

lim（x趋近于0）（cos2x-cos3x）/x²

Answer 1

答：5/2
用洛必达法则：
lim(x-->0) (cos2x - cos3x)/x²
= lim(x-->0) (- 2sin2x + 3sin3x)/(2x)
= lim(x-->0) (- 4cos2x + 9cos3x)/(2)
= (- 4 + 9)/2 = 5/2
用和差化积公式：
lim(x-->0) (cos2x - cos3x)/x²
= lim(x-->0) - 2[sin(5x/2)sin(- x/2)]/x²
= lim(x-->0) sin(5x/2)/x * 2sin(x/2)/x
= lim(x-->0) [sin(5x/2)/(5x/2)](5/2) * [sin(x/2)/(x/2)]
= (1)(5/2)(1)
= 5/2