Cool Limits!!

$\large{\displaystyle \lim_{x\to 0} \frac{(2016x)!-1}{(2015x)!-1}}$

Applying L-Hospital

$\large{\displaystyle \underset { x\rightarrow 0 }{ \lim }\frac{2016}{2015} \times \frac { { \Gamma }^{ \prime }\left( 2016x+1 \right) }{ { \Gamma }^{ \prime }\left( 2015x+1 \right) } \qquad \left[ \because \quad \Gamma \left( x+1 \right) =x! \right] }$

we know $\large{\psi \left( x \right) =\frac { { \Gamma }^{ \prime }\left( x \right) }{ \Gamma \left( x \right) } }$

where $\psi (x)$ is digamma function and $\Gamma (x)$ is gamma function.

$\large{\frac{2016}{2015}\displaystyle \lim_{x\to 0} \frac { \psi \left( 2016x+1 \right) \Gamma \left( 2016x+1 \right) }{ \psi \left( 2015x+1 \right) \Gamma \left( 2015x+1 \right) } }$

After applying the limit

we get $\large{\boxed{\frac{2016}{2015}}}$

$\large{10^4 \times \frac{2016}{2015}=10004.9}$

$\begin{array} {rll} L & = \displaystyle \lim_{x \to 0} \dfrac{(2016x)!-1}{(2015x)!-1} & \small \color{#3D99F6}{x! = \Gamma(x+1)} \\ & = \displaystyle \lim_{x \to 0} \dfrac{\Gamma(2016x+1)-1}{\Gamma(2015x+1)-1} & \small \color{#3D99F6}{\text{Since } L \to \frac{0}{0} \text{, we can use l'Hôpital's rule.}} \\ & = \displaystyle \lim_{x \to 0} \dfrac{\frac{d}{dx} [\Gamma (2016x+1)-1]}{\frac{d}{dx} [\Gamma (2015x+1)-1]} \\ & = \displaystyle \lim_{x \to 0} \dfrac{2016\Gamma (2016x+1) \psi_0 (2016x+1)}{2015\Gamma (2015x+1) \psi_0 (2015x+1)} \\ & = \dfrac{2016\Gamma (1) \psi_0 (1)}{2015\Gamma (1) \psi_0 (1)} \\ & = \dfrac{2016}{2015} \\ & \\ \Rightarrow \lfloor 10000L \rfloor & = \boxed{10004} \end{array}$