Question

The value of $\underset{x\to 0}{\lim }\frac{x^{6000}-{\left(sinx\right)}^{6000}}{x^2{\left(sinx\right)}^{6000}}$ is

• A. $1000$
• B. $100$
• C. $1100$
• D. $1010$

Answer 1

Answer: (A)

$\underset{x\to 0}{\lim }\frac{x^{6000}-{\left(sinx\right)}^{6000}}{x^2\frac{{\left(sinx\right)}^{6000}}{x^{6000}}.x^{6000}}\left(\frac{0}{0}\right)$
$\underset{x\to 0}{\lim }\frac{x^{6000}-{\left(sinx\right)}^{6000}}{x^{6002}}$
$=-\underset{x\to 0}{\lim }\frac{{\left(sinx\right)}^{6000}-x^{6000}}{x^{6002}}$
$=-\underset{x\to 0}{\lim }\frac{{\left(\frac{sinx}{x}\right)}^{6000}-1}{x^2}$
$=-\underset{x\to 0}{\lim }\frac{6000{\left(\frac{sinx}{x}\right)}^{5999}\left(\frac{xcosx-sinx}{x^2}\right)}{2x}$ (By L-Hospital Rule)
$=-\frac{6000}{2}\underset{x\to 0}{\lim }\frac{xcosx-sinx}{x^3}$

$=-3000\underset{x\to 0}{\lim }\frac{cosx-xsinx-cosx}{3x^2}$
$=1000$