Question

$\underset{x\to 0}{lim}\frac{{\tan }^{2}x-x^2}{x^2{\tan }^{2}x}$

Answer 1

$\underset{x\to 0}{lim}\frac{ta{n}^{2}x-x^2}{x^{2}ta{n}^{2}x}$

$=\underset{x\to 0}{lim}\frac{\left(\frac{ta{n}^{2}x}{x^2}\right)\times x^2-x^2}{x^2\times \left(\frac{ta{n}^{2}x}{x^2}\right)\times x^2}$

$=\underset{x\to 0}{lim}\frac{0}{x^4}$ $(\underset{x\to 0}{lim}\frac{\tan x}{x}=1)$

Applying L-hospital rule

$=\underset{x\to 0}{lim}\frac{0}{4x^3}$

Again applying L-hospital rule

$=\underset{x\to 0}{lim}\frac{0}{12x^2}$

Again applying L-hospital rule

$=\underset{x\to 0}{lim}\frac{0}{24x}$

Again applying L-hospital rule

$=\underset{x\to 0}{lim}\frac{0}{24}$

$=0$