$\lim _{x \rightarrow \infty} \frac{(2 x+1)^{50}+(2 x+2)^{50}+(2 x+3)^{50}+\cdots \cdots+(2 x+100)^{50}}{(2 x)^{50}+(10)^{50}}=\cdots$

The limit $\lim _{x \rightarrow \infty} \frac{(2 x+1)^{50}+(2 x+2)^{50}+\cdots+(2 x+100)^{50}}{(2 x)^{50}+(10)^{50}}$ is evaluated by identifying dominant terms as $x$ approaches infinity.

Factor $(2x)^{50}$ from each term in the numerator: $(2x+k)^{50} = (2x)^{50}\left(1 + \frac{k}{2x}\right)^{50}$ for $k = 1$ to $100$.

The denominator simplifies to $(2x)^{50}\left(1 + \frac{10^{50}}{(2x)^{50}}\right)$, with $(2x)^{50}$ dominant as $x \to \infty$.

Dividing numerator and denominator by $(2x)^{50}$ yields $\frac{\sum_{k=1}^{100} \left(1 + \frac{k}{2x}\right)^{50}}{1 + \frac{10^{50}}{(2x)^{50}}}$.

As $x \to \infty$, each $\left(1 + \frac{k}{2x}\right)^{50} \to 1$ and $\frac{10^{50}}{(2x)^{50}} \to 0$, so the limit simplifies to $\frac{100 \cdot 1}{1} = 100$.

Answer: $\boxed{100}$