First slide

Introduction to limits

Question

The value of $lim_{x \to 1} \frac{\log x}{\sin π x},$ is

Moderate

A

$\frac{1}{π}$
B

$- π$
C

$π$
D

$- \frac{1}{π}$

Solution

We have,
$\begin{array}{l} lim_{x \to 1} \frac{\log x}{\sin π x} = lim_{x \to 1} \frac{\log [1 + (x - 1)}}{\sin (π - π x)} \\ = lim_{x \to 1} \frac{\log {1 + (x - 1)}}{\sin π (1 - x)} \\ = lim_{x \to 1} \frac{\log {1 + (x - 1)}}{x - 1} \times \frac{x - 1}{\sin π (1 - x)} \\ = - \frac{1}{π} lim_{x \to 1} \frac{\log {1 + (x - 1)}}{x - 1} \times - \frac{π (1 - x)}{\sin π (1 - x)} = - \frac{1}{π} \end{array}$

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