$\underset{x\to \infty }{\lim }\frac{(x+1{)}^{10}+(x+2{)}^{10}+\cdots +(x+100{)}^{10}}{x^{10}+{10}^{10}}$  is equal to

$f\left(x\right)=\frac{(x+1{)}^{10}+(x+2{)}^{10}+\cdots +(x+100{)}^{10}}{x^{10}+{10}^{10}}$

$=\frac{{\left(1+\frac{1}{x}\right)}^{10}+{\left(1+\frac{2}{x}\right)}^{10}+\cdots +{\left(1+\frac{100}{x}\right)}^{10}}{1+{\left(\frac{10}{x}\right)}^{10}}$

As $\frac{1}{x}\to 0,\text{ for }x\to \infty \text{ so }\underset{x\to \infty }{\lim }f\left(x\right)=\frac{1+1\cdots +1-\left(100\text{ times }\right)}{1}$

                                                         $=100$.