Write the Value of Lim X → ∞ Sin X X . - Mathematics

Write the value of $\lim_{x \to \infty} \frac{\sin x}{x} .$

Solution

$\lim_{x \to \infty} \left( \frac{\sin x}{x} \right)$
$\text{ Let } x = \frac{1}{y}$
$\text{ If } x \to \infty , \text{ then } y \to 0 .$
$= \lim_{y \to 0} y \cdot \sin \left( \frac{1}{y} \right)$
$\text{ LHL }:$
$\text{ Let } y = 0 - h$
$\text{ If } y \to 0, \text{ then } h \to 0 .$
$= \lim_{h \to 0} \left( \left( 0 - h \right) \times \sin \left( \frac{1}{0 - h} \right) \right)$
$0 \text{ times The oscillating number between } -1 \text{ and } 1$
$= 0$
$\text{ RHL }:$
$\lim_{y \to 0^+} \left( y \cdot \sin \left( \frac{1}{y} \right) \right)$
$Let y = 0 + h$
$\text{ If } y \to 0, \text{ then } h \to 0 .$
$= \lim_{h \to 0} h \times \sin \left( \frac{1}{h} \right)$
$= 0 \text{ times The oscillating number between } -1 \text{ and } 1$
$= 0$

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Q 7 | Page 77