Answer to Question #314383 in Real Analysis for Pankaj

Question

Find the following limit.:

Lim x →0 for 1-cos x²/x².sin x²

Answer 1

$\lim\limits _{x\to 0}\frac{1-cosx^2}{x^2sinx^2}=\\ \colorbox{aqua}{We use formulas}\\ 1-cosa=2sin^2\frac{a}{2}\\ sina=2sin\frac{a}{2}cos\frac{a}{2}\\ =\lim\limits _{x\to 0}\frac{2sin^2\frac{x^2}{2}}{x^22sin\frac{x^2}{2}cos\frac{x^2}{2}}=\lim\limits _{x\to 0}\frac{sin\frac{x^2}{2}}{x^2cos\frac{x^2}{2}}=\\ \colorbox{aqua}{We use formulas}\\ \lim\limits _{x\to 0}\frac{sinx}{x}=1\\ =\lim\limits _{x\to 0}\frac{sin\frac{x^2}{2}}{2\cdot\frac{x^2}{2}cos\frac{x^2}{2}}= \frac{1}{2cos0}=\frac{1}{2}$