Answer to Question #205465 in Real Analysis for Aman

Question

Find lim x tends to 0 1-cos²x /x²sinx²

Answer 1

$$\lim\limits_{x\to 0}\dfrac{1-\cos^2(x)}{x^2\sin(x^2)}=\lim\limits_{x\to 0}\dfrac{\sin^2(x)}{x^2\sin(x^2)}$$

$$=\lim\limits_{x\to 0}\bigg((\dfrac{\sin(x)}{x})^2\cdot\dfrac{1}{\sin(x^2)}\bigg)$$

$$=(1)^2\lim\limits_{x\to 0}(\dfrac{1}{\sin(x^2)})=\infin$$