Question

Match the following
Column I
([.] denotes the greatest integer function)

Answer 1

We know that $\underset{x\to 0}{lim}\frac{\sin x}{x}=1$ (but a value which is smaller than $1$)
or $\left[\underset{x\to 0}{lim}100\frac{\sin x}{x}\right]=99$

and $\left[\underset{x\to 0}{lim}100\frac{x}{\sin x}\right]=100$

Also, $\underset{x\to 0}{lim}\frac{{\sin }^{-1}x}{x}=1$ (but a value which is more than 1)

or $\left[\underset{x\to }{lim}100\frac{{\sin }^{-1}x}{x}\right]=100$

and $\left[\underset{x\to }{lim}100\frac{x}{{\sin }^{-1}x}\right]=99$

$\underset{x\to 0}{lim}\frac{\tan x}{x}=1$ (but a value which is bigger than $1$)

or $\left[\underset{x\to 0}{lim}100\frac{\tan x}{x}\right]=100$

and $\left[\underset{x\to 0}{lim}100\frac{{\tan }^{-1}x}{x}\right]=99$
Hence,

A.
$\underset{x\to 0}{lim}(\left[100\frac{\sin x}{x}\right]+\left[100\frac{\tan x}{x}\right])=199$

B.
$\underset{x\to 0}{lim}(\left[100\frac{x}{\sin x}\right]+\left[100\frac{\tan x}{x}\right])=200$

C.

$\underset{x\to 0}{lim}(\left[100\frac{{\sin }^{-1}x}{x}\right]+\left[100\frac{{\tan }^{-1}x}{x}\right])=199$

D.

$\underset{x\to 0}{lim}(\left[100\frac{x}{{\sin }^{-1}x}\right]+\left[100\frac{{\tan }^{-1}x}{x}\right])=198$