Question

${lim}_{x\to 0}(\frac{1}{si{n}^{2}x}-\frac{1}{sin{h}^{2}x})$

Answer 1

$L=\underset{x\to 0}{lim}(\frac{1}{{\sin }^{2}x}-\frac{1}{{\sin }^{2}x})$

$=\underset{x\to 0}{lim}\left(\frac{\sin h^{2}x-{\sin }^{2}x}{{\sin }^{2}x\sin h^{2}x}\right)$

$=\underset{x\to 0}{lim}\frac{{(x+\frac{x^3}{3!}+\frac{x^5}{5!}+....)}^2-{(x-\frac{x^3}{3!}+\frac{x^5}{5!}-....)}^2}{{(x-\frac{x^3}{3!}+\frac{x^5}{5!}....)}^2{(x+\frac{x3}{3!}+\frac{x^5}{5!}+....)}^2}$

$L=\underset{x\to 0}{lim}2.\frac{2x^4}{3!x^4}$

$=\frac{4}{3!}$

$\therefore L=\frac{4}{6}$

$\therefore L=\frac{2}{3}$