Question

Let $f$ be a differentiable function satisfying $\underset{0}{\overset{f(x)}{\int }}{f}^{-1}(t)dt-\underset{0}{\overset{x}{\int }}(\cos t-f(t))dt=0$ and $f(0)=1$
The value of $\underset{x\to 0}{\mathrm{Lim}}\left(\left[\frac{\cos x}{f(x)}\right]+\left[\frac{\cos 2x}{f(2x)}\right]+\left[\frac{\cos 3x}{f(3x)}\right]+\dots \dots +\left[\frac{\cos (100x)}{f(100x)}\right]\right)$ is equal to [Note : where $[k]$ denotes greatest integer less than or equal to $k$.]

• A. 0
• B. 4950
• C. 5049
• D. 5050

Answer 1

Answer: (A)

$\underset{x\to 0}{\mathrm{Lim}}\left(\left[\frac{\cos x}{f(x)}\right]+\left[\frac{\cos 2x}{f(2x)}\right]+\dots \dots +\left[\frac{\cos (100x)}{f(100x)}\right]\right)$
$\underset{x\to 0}{\mathrm{Lim}}\left(\left[\frac{x}{\tan x}\right]+\left[\frac{2x}{\tan 2x}\right]+\dots \dots +\left[\frac{100x}{\tan 100x}\right]\right)=0$