First slide

Introduction to limits

Question

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

Moderate

A

0
B

1
C

10
D

100

Solution

$\begin{array}{l} lim_{x \to \infty} \frac{(x + 1)^{10} + (x + 2)^{10} + \dots + (x + 100)^{10}}{x^{10} + 10^{10}} \\ = lim_{x \to \infty} \frac{x^{10} \cdot [{(1 + \frac{1}{x})}^{10} + {(1 + \frac{2}{x})}^{10} + \dots + {(1 + \frac{100}{x})}^{10}]}{x^{10} [1 + \frac{10^{10}}{x^{10}}]} \\ = lim_{x \to \infty} \frac{[{(1 + \frac{1}{x})}^{10} + {(1 + \frac{2}{x})}^{10} + \dots + {(1 + \frac{100}{x})}^{10}]}{[1 + \frac{10^{10}}{x^{10}}]} = 100 \end{array}$

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