How can I evaluate $\lim_{x \to \infty}\frac{\sum_{k=1}^{\ 1000}(x+k)^{10}}{x^{10}+10^{10}}$? I would like to examine the limit of the following function as x goes towards infinity:
$$\lim_{x \to \infty}\frac{\sum\limits_{k=1}^{\ 1000}(x+k)^{10}}{x^{10}+10^{10}}$$ 
I already have tried to split the sum, exlude the factor $\ x^{10} $ , then reduce the fraction by it in the numerator and denumerator, so that i might see a way to determine the limit. But that does not seem to help me proceeding
And using l’Hospital´s Rule seems to be too complicated.

Here the "honest" calculation using properties of limits:
\begin{eqnarray*} \frac{\sum_{k=1}^{\ 1000}(x+k)^{10}}{x^{10}+10^{10}}
& = & \frac{\sum_{k=1}^{\ 1000}\left(1+\frac{k}{x}\right)^{10}}{1+\frac{10^{10}}{x^{10}}}\\
& \stackrel{x\to \infty}{\longrightarrow} & \frac{\sum_{k=1}^{\ 1000}\left(1+0\right)^{10}}{1+0} \\
& = & \sum_{k=1}^{\ 1000} 1 = 1000
\end{eqnarray*}
