lim(x趋于0)(x^2-sin^2xcos^2x)/((x^2)sin^2x)

Question

lim(x趋于0)(x^2-sin^2xcos^2x)/((x^2)sin^2x)

Answer 1

lim(x→0) (x² - sin²xcos²x)/(x²sin²x)
= lim(x→0) [x² - (1/2 * sin2x)²]/x⁴、sinx ~ x
= lim(x→0) [x² - (1/4)(1 - cos4x)/2]/x⁴
= (1/8)lim(x→0) (8x² - 1 + cos4x)/x⁴、洛必达法则
= (1/8)lim(x→0) (16x - 4sin4x)/(4x³)
= (1/8)lim(x→0) (4x - sin4x)/x³、洛必达法则
= (1/8)lim(x→0) (4 - 4cos4x)/(3x²)
= (1/8)lim(x→0) (16sin4x)/(6x)
= (1/8)(16/6)(4)
= 4/3