Answer to Question #200203 in Real Analysis for Rajkumar

Question

Find lim x→0 (1-cos²x) / (x² sinx²)

Answer 1

Let us find the limit $\lim\limits_{ x\to 0} \frac{1-\cos^2x}{ x^2 \sin x^2}:$

$\lim\limits_{ x\to 0} \frac{1-\cos^2x}{ x^2 \sin x^2}= \lim\limits_{ x\to 0} \frac{\sin^2x}{ x^2 \sin x^2}= \lim\limits_{ x\to 0} \frac{\sin x}{ x}\cdot \lim\limits_{ x\to 0} \frac{\sin x}{ x}\cdot \lim\limits_{ x\to 0} \frac{1}{ \sin x^2}=1\cdot 1\cdot \infty=\infty.$