# Lim X → 1 1 − 1 X Sin π ( X − 1 ) - Mathematics

$\lim_{x \to 1} \frac{1 - \frac{1}{x}}{\sin \pi \left( x - 1 \right)}$

#### Solution

$\lim_{x \to 1} \left[ \frac{1 - \frac{1}{x}}{sin\pi\left( x - 1 \right)} \right]$
$= \lim_{x \to 1} \left[ \frac{x - 1}{xsin\pi\left( x - 1 \right)} \right]$

Let y = x – 1
If x → 1, then y → 0.

$= \lim_{y \to 0} \left[ \frac{y}{\left( y + 1 \right)sin\pi y} \right]$

$= \lim_{y \to 0} \left[ \frac{1}{\pi\left( y + 1 \right) \frac{\sin \pi y}{\pi y}} \right]$

$= \frac{1}{\pi\left( 0 + 1 \right) \times 1}$

$= \frac{1}{\pi}$

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.8 | Q 23 | Page 62