First slide

Introduction to limits

Question

The value of the limit $lim_{x \to 0} {\{1^{1 / \sin^{2} x} + 2^{1 / \sin^{2} x} + \dots + n^{1 / \sin^{2} x}\}}^{\sin^{2} x}$ is

Moderate

A

$\infty$
B

0
C

$\frac{n (n + 1)}{2}$
D

n

Solution

$lim_{x \to 0} {\{1^{1 / \sin^{2} x} + 2^{1 / \sin^{2} x} + \dots + n^{1 / \sin^{2} x}\}}^{\sin^{2} x}$
put $\frac{1}{\sin^{2} x} = t \geq 1$
$\begin{array}{l} ∴ lim_{t \to \infty} {(1^{t} + 2^{t} + \dots + n^{t})}^{1 / t} \\ = lim_{t \to \infty} {(n^{t})}^{1 / t} {[{(\frac{1}{n})}^{t} + {(\frac{2}{n})}^{t} + \dots + 1]}^{1 / t} \\ = n lim_{t \to \infty} {[{(\frac{1}{n})}^{t} + {(\frac{2}{n})}^{t} + \dots + 1]}^{1 / t} \\ = n [0 + 0 + \dots + 1]^{0} = n \end{array}$

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