Question

The value of $\underset{x\to 0}{\lim }\frac{1}{x^{18}}\left(1-\cos \left(\frac{x^3}{3}\right)-\cos \left(\frac{x^6}{6}\right)+\cos \left(\frac{x^3}{3}\right)\cdot \cos \left(\frac{x^6}{6}\right)\right)$ is ${\lambda }^2$, then the value of $900\lambda$ is equal to (here, $\lambda >0$ )

Answer 1

Answer: 25

$\underset{x\to 0}{\lim }\left\{1-\cos \left(\frac{x^3}{3}\right)-\cos \left(\frac{x^6}{6}\right)\left(1-\cos \left(\frac{x^3}{3}\right)\right)\right\}\cdot \frac{1}{x^{18}}$
$\underset{x\to 0}{\lim }\frac{\left(1-\cos \left(\frac{x^3}{3}\right)\right)\left(1-\cos \left(\frac{x^6}{6}\right)\right)}{x^{18}}$
$\underset{x\to 0}{\lim }\frac{1-\cos \left(\frac{x^3}{3}\right)}{{\left(\frac{x^3}{3}\right)}^2}\left(\frac{x^6}{9}\right)\cdot \frac{1-\cos \left(\frac{x^6}{6}\right)}{{\left(\frac{x^6}{6}\right)}^2}\cdot {\left(\frac{x^6}{6}\right)}^2\cdot \frac{1}{x^{18}}$
$\underset{x\to 0}{\lim }\frac{1}{2}\times \frac{x^6}{9}\cdot \frac{1}{2}\cdot \frac{x^{12}}{36}\cdot \frac{1}{x^{18}}=\frac{1}{(36{)}^2}$
$\therefore \lambda =\frac{1}{36}$