Question

$\underset{\theta \to 0}{lim}\left(\left[\frac{n\sin \theta }{\theta }\right]+\left[\frac{n\tan \theta }{\theta }\right]\right)$, where [.] represent greatest integer function and $n\in N$ is equal to

• A. $2n$
• B. $2n+1$
• C. $2n-1$
• D. does not exist

Answer 1

The correct option is A $2n$

$[.]$ gives an integer as a end result we can insert the limit function inside without any change in the final result.

and as we know

${lim}_{x\to 0}\frac{sin\theta }{\theta }=1$

${lim}_{x\to 0}\frac{ntan\theta }{\theta }=1$

Hence,

$\left[{lim}_{x\to 0}\frac{nsin\theta }{\theta }\right]=\left[{lim}_{x\to 0}\frac{ntan\theta }{\theta }\right]=\left[n\right]$

as $nϵN,\left[n\right]=n$

Hence,

$n+n=2n$