First slide

Introduction to limits

Question

$lim_{n \to 0} \frac{(1 - \frac{1}{2^{2}}) (1 - \frac{1}{3^{2}}) \dots (1 - \frac{1}{n^{2}})}{(1 - \frac{1}{2}) (1 - \frac{1}{3}) \dots (1 - \frac{1}{n})}$ is equal to

Moderate

A

$- \frac{1}{2}$
B

$\frac{1}{2}$
C

2
D

-2

Solution

$\begin{array}{l} lim_{n \to 0} \frac{(1 - \frac{1}{2^{2}}) (1 - \frac{1}{3^{2}}) \dots (1 - \frac{1}{n^{2}})}{(1 - \frac{1}{2}) (1 - \frac{1}{3}) \dots (1 - \frac{1}{n})} \\ = lim_{n \to 0} \frac{(1 + \frac{1}{2}) (1 + \frac{1}{3}) \dots (1 + \frac{1}{n}) (1 - \frac{1}{2}) (1 - \frac{1}{3}) \dots (1 - \frac{1}{n})}{(1 - \frac{1}{2}) (1 - \frac{1}{3}) \dots (1 - \frac{1}{n})} \\ = lim_{n \to 0} \frac{3}{2} \times \frac{4}{3} \times \dots \times \frac{n + 1}{n} = lim_{n \to 0} \frac{n + 1}{2} = \frac{1}{2} \end{array}$

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