Question

Let $f$ be a differentiable function satisfying $\int\limits_0^{f(x)} f^{-1}(t) d t-\int\limits_0^x(\cos t-f(t)) d t=0$ and $f(0)=1$
The value of $\underset {x \rightarrow 0}{\text{Lim}}\left(\left[\frac{\cos x}{f(x)}\right]+\left[\frac{\cos 2 x}{f(2 x)}\right]+\left[\frac{\cos 3 x}{f(3 x)}\right]+\ldots \ldots+\left[\frac{\cos (100 x)}{f(100 x)}\right]\right)$ 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 \rightarrow 0}{\text{Lim}}\left(\left[\frac{\cos x}{f(x)}\right]+\left[\frac{\cos 2 x}{f(2 x)}\right]+\ldots \ldots+\left[\frac{\cos (100 x)}{f(100 x)}\right]\right) $
$\underset {x \rightarrow 0}{\text{Lim}}\left(\left[\frac{x}{\tan x}\right]+\left[\frac{2 x}{\tan 2 x}\right]+\ldots \ldots+\left[\frac{100 x}{\tan 100 x}\right]\right)=0 $