Question

The value of $\underset{x\to \infty }{\lim }\frac{{\left(x+1\right)}^{10}+{\left(x+2\right)}^{10}+...+{\left(x+100\right)}^{10}}{x^{10}+{10}^{10}}$ is

(a) 10

(b) 100

(c) 10¹⁰

(d) none of these

Answer 1

(b) 100

$$\begin{gathered} \underset{x\to \infty }{\lim }\frac{{\left(x+1\right)}^{10}+{\left(x+2\right)}^{10}+....+{\left(x+100\right)}^{10}}{x^{10}+{10}^{10}} \\ \mathrm{Dividing}{\mathrm{N}}^r\mathrm{and}{\mathrm{D}}^r\mathrm{by}{x}^{10}: \\ \Rightarrow \underset{x\to \infty }{\lim }\frac{{\left(1+{\displaystyle \frac{1}{x}}\right)}^{10}+{\left(1+{\displaystyle \frac{2}{x}}\right)}^{10}+....+{\left(1+{\displaystyle \frac{100}{x}}\right)}^{10}}{1+{\displaystyle \frac{{10}^{10}}{x^{10}}}} \\ =1+1+1+...+100\mathrm{times} \\ =100 \end{gathered}$$