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  4. displaystyle lim x arrow ∞ ((x+1)10+(x+2)10+ ldots .+(x+100)10/x10+1010) is equal to

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Q. $\displaystyle\lim _{x \rightarrow \infty} \frac{(x+1)^{10}+(x+2)^{10}+\ldots .+(x+100)^{10}}{x^{10}+10^{10}}$ is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow \infty} \frac{(x+1)^{10}+(x+2)^{10}+\ldots .+(x+100)^{10}}{x^{10}+10^{10}}$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{x^{10}\left[\left(1+\frac{1}{x}\right)^{10}+\left(1+\frac{2}{x}\right)^{10}+\ldots+\left(1+\frac{100}{x}\right)^{10}\right]}{x^{10}\left[1+\frac{10^{10}}{x^{10}}\right]}=100.$