_ x (x + 1) ^ 10 + (x + 2) ^ 10 + ..... + (x + 100) ^ 10 x^ 10 + … - Vidyadip

Question

$\mathop {\lim }\limits_{x \to \infty } \frac{{{{(x + 1)}^{10}} + {{(x + 2)}^{10}} + ..... + {{(x + 100)}^{10}}}}{{{x^{10}} + {{10}^{10}}}}$ is equal to

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Answer

d
(d) $\mathop {\lim }\limits_{x \to \infty } \frac{{{{(x + 1)}^{10}} + {{(x + 2)}^{10}} + ...... + {{(x + 100)}^{10}}}}{{{x^{10}} + {{10}^{10}}}}$

$ = \mathop {\lim }\limits_{x \to \infty } \frac{{{x^{10}}\left[ {{{\left( {1 + \frac{1}{x}} \right)}^{10}} + {{\left( {1 + \frac{2}{x}} \right)}^{10}} + ... + {{\left( {1 + \frac{{100}}{x}} \right)}^{10}}} \right]}}{{{x^{10}}\left[ {1 + \frac{{{{10}^{10}}}}{{{x^{10}}}}} \right]}} = 100$.

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