Q 73. - Calculus

Question

Calculate the limit.

$\underset{x\to 0^{+}}{\lim } \frac{x^{2}csc 3x}{1-\cos 2x}$

Step-by-Step Solution

Answer

Answer is $\infty$

Step 1. Given information

Limit function is $\underset{x\to 0^{+}}{\lim } \frac{x^{2}csc 3x}{1-\cos 2x}$

Step 2. Explanation

$\begin{array}{rcl}\underset{x\to 0^{+}}{\lim } \frac{x^{2}csc 3x}{1-\cos 2x}& =& \underset{x\to 0^{+}}{\lim }\frac{x^2}{\sin 3x}\left(\frac{1}{1-\left(1-2{\sin }^{2}x\right)}\right)\\ & =& \underset{x\to 0^{+}}{\lim }\frac{1}{{\displaystyle \frac{\sin 3x}{3x}}\left(3x\right)}\left(\frac{1}{2{\displaystyle \frac{{\sin }^{2}x}{x^2}}}\right)\\ & =& \underset{x\to 0^{+}}{\lim }\frac{1}{6x}\\ & =& \infty \\ & & \end{array}$