lim-x-0-sin-2-x-sin-x-2-x-2-cos-2-x-cos-x-2-

Question Number 197282 by cortano12 last updated on 12/Sep/23

\lim_{x \to 0} \frac{\sin^{2} x - \sin x^{2}}{x^{2} (\cos^{2} x - \cos x^{2})} = ?

Answered by MM42 last updated on 12/Sep/23

{l i m}_{x \to 0} \frac{{s i n}^{2} x - x^{2} + x^{2} - {s i n x}^{2}}{x^{2} ({c o s}^{2} x - 1 + 1 - {c o s x}^{2})}

= {l i m}_{x \to 0} \frac{(s i n x + x) (s i n x - x) + \frac{1}{6} x^{6}}{x^{2} ((c o s x - 1) (c o s x + 1) + \frac{1}{2} x^{4})}

= {l i m}_{x \to 0} \frac{(2 x) (- \frac{1}{6} x^{3}) + \frac{1}{6} x^{6}}{x^{2} ((- \frac{1}{2} x^{2}) (c o s x + 1) + \frac{1}{2} x^{4})}

= {l i m}_{x \to 0} \frac{- \frac{1}{3} x^{4} + \frac{1}{6} x^{6}}{- x^{4} + \frac{1}{2} x^{6}} = \frac{1}{3} ✓

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