sin(x)=x-x³/3!+x⁵/5! approx.
x-sin(x)=x³/3!-x⁵/5! approx.
x+sin(x)=2x-x³/3!+x⁵/5! approx.
x²-sin²(x)=(x-sin(x))(x+sin(x))=(x³/3!-x⁵/5!)(2x-x³/3!+x⁵/5!).
x²-sin²(x)=x⁴/3–x⁶/36 if we ignore powers of x greater than x⁶.
xsin(x)=x²-x⁴/3!+x⁶/5!=x²(1-x²/3!+x⁴/5!) approx.
(xsin(x))²=x⁴(1-x²/3+x⁴/36+x⁸/14400+2x⁴/5!(1-x²/3!)).
(xsin(x))²=x⁴-x⁶/3+x⁸/36+x¹²/14400+2x⁸/5!(1-x²/3!).
When we ignore powers of x greater than x⁶, we get:
(xsin(x))²=x⁴-x⁶/3.
So (x²-sin²(x))/(x²sin²(x))=(x⁴/3–x⁶/36)/(x⁴-x⁶/3).
Divide top and bottom by common factor x⁴:
(1/3-x²/36)/(1-x²/3).
As x→0, this expression approaches ⅓.
Therefore the limit is ⅓.