Question

Solve the limit:

$\boxed{\lim \limits_{x \to \infty} \left[ \dfrac{ {(x + 1)}^{10} + {(x + 2)}^{10} + ( {x + 3)}^{10} + ... + {(x + 100)}^{10}}{ {x}^{10} + {10}^{10} } \right]}$​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\displaystyle\lim_{x\to\infty}\,\dfrac{\left(x+1\right)^{10}+\left(x+2\right)^{10}+\left(x+3\right)^{10}+\cdots+\left(x+100\right)^{10}}{{x}^{10}+{10}^{10}}$

$\displaystyle=\lim_{x\to\infty}\,\dfrac{{x}^{10}\left(1+\dfrac{1}{x}\right)^{10}+{x}^{10}\left(1+\dfrac{2}{x}\right)^{10}+{x}^{10}\left(1+\dfrac{3}{x}\right)^{10}+\cdots+{x}^{10}\left(1+\dfrac{100}{x}\right)^{10}}{{x}^{10}\left(1+\dfrac{{10}^{10}}{{x}^{10}}\right)}$

$\displaystyle=\lim_{x\to\infty}\,\dfrac{{x}^{10}\left\{\left(1+\dfrac{1}{x}\right)^{10}+\left(1+\dfrac{2}{x}\right)^{10}+\left(1+\dfrac{3}{x}\right)^{10}+\cdots+\left(1+\dfrac{100}{x}\right)^{10}\right\}}{{x}^{10}\left(1+\dfrac{{10}^{10}}{{x}^{10}}\right)}$

$\displaystyle=\lim_{x\to\infty}\,\dfrac{\left(1+\dfrac{1}{x}\right)^{10}+\left(1+\dfrac{2}{x}\right)^{10}+\left(1+\dfrac{3}{x}\right)^{10}+\cdots+\left(1+\dfrac{100}{x}\right)^{10}}{1+\dfrac{{10}^{10}}{{x}^{10}}}$

$=\dfrac{\left(1+0\right)^{10}+\left(1+0\right)^{10}+\left(1+0\right)^{10}+\cdots+\left(1+0\right)^{10}}{1+0}$

$=\dfrac{100}{1}$

$=100$