mathop {lim }limits_{x to infty } frac{{{{(x | vedclass

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(D) We are given the limit: $\mathop {\lim }\limits_{x 	o \infty } \frac{{{{(x + 1)}^{10}} + {{(x + 2)}^{10}} + \dots + {{(x + 100)}^{10}}}}{{{x^{10}

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$100$

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(D) We are given the limit: $\mathop {\lim }\limits_{x 	o \infty } \frac{{{{(x + 1)}^{10}} + {{(x + 2)}^{10}} + \dots + {{(x + 100)}^{10}}}}{{{x^{10}} + {{10}^{10}}}}$Divide the numerator and denominator by $x^{10}$:$= \mathop {\lim }\limits_{x 	o \infty } \frac{{{{\left( {1 + \frac{1}{x}} ight)}^{10}} + {{\left( {1 + \frac{2}{x}} ight)}^{10}} + \dots + {{\left( {1 + \frac{{100}}{x}} ight)}^{10}}}}{{1 + \frac{{{{10}^{10}}}}{{{x^{10}}}}}}$As $x 	o \infty$,the terms $\frac{k}{x} 	o 0$ for any constant $k$,and $\frac{10^{10}}{x^{10}} 	o 0$.Therefore,the expression becomes:$= \frac{{{{(1 + 0)}^{10}} + {{(1 + 0)}^{10}} + \dots + {{(1 + 0)}^{10}}}}{{1 + 0}}$Since there are $100$ terms in the numerator,each equal to $1^{10} = 1$:$= \frac{1 + 1 + \dots + 1 	ext{ (100 times)}}{1} = 100$.

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

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