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Question
The value of \[\lim_{x \to \infty} \frac{\left( x + 1 \right)^{10} + \left( x + 2 \right)^{10} + . . . + \left( x + 100 \right)^{10}}{x^{10} + {10}^{10}}\] is
Options
10
100
1010
none of these
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Solution
\[\lim_{x \to \infty} \frac{\left( x + 1 \right)^{10} + \left( x + 2 \right)^{10} + . . . . + \left( x + 100 \right)^{10}}{x^{10} + {10}^{10}}\]
\[\text{ Dividing } N^r \text{ and } D^r \text{ by } x^{10} : \]
\[ \Rightarrow \lim_{x \to \infty} \frac{\left( 1 + \frac{1}{x} \right)^{10} + \left( 1 + \frac{2}{x} \right)^{10} + . . . . + \left( 1 + \frac{100}{x} \right)^{10}}{1 + \frac{{10}^{10}}{x^{10}}}\]
\[ = 1 + 1 + 1 + . . . + 100 \text{ times }\]
\[ = 100\]
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