\(\lim\limits_{x \to 10} \) \(\cfrac{cos9x-cos7x}{cos3x-cos5x}\) (\(\frac{0}{0}\)type)
= \(\lim\limits_{x \to 10} \) \(\cfrac{-9sin9x +7sin7x}{-3sin3x+5sinx5x}\) (By applying D.L.H Rule)
= \(\lim\limits_{x \to 10} \) \(\cfrac{-9\frac{sin9x}{9x}\times9x+7\frac{sin7x}{7x}\times7x}{-3\frac{sin3x}{3x}\times3x+5\frac{sin5x}{5x}\times5x}\)
= \(\lim\limits_{x \to 10} \) \(\cfrac{x\Big(-81\frac{sin9x}{9x}+49\frac{sin7x}{7x}\Big)}{x\Big(-9\frac{sin3x}{3x}+25\frac{sin5x}{5x}\Big)}\)
= \(\cfrac{-81\lim\limits_{x \to 10}\frac{sin9x}{9x}+49\lim\limits_{x \to10}\frac{sin7x}{7x}}{-9\lim\limits_{x \to 10}\frac{sin3x}{3x}+25\lim\limits_{x \to 10}\frac{sin5x}{5x}}\)
= \(\frac{-81+49}{-91+25}\) = \(\frac{-32}{16}\) = -2 (∵\(\lim\limits_{x \to 10}\) \(\frac{sinax}{ax}\) = 1)