Math Doubts

$\displaystyle \lim_{x \to 0} \dfrac{\sin{x}}{x}$ Proof in General Method

$x$ is a literal but represents angle of a right angled triangle and the sine function is represented by $\sin{x}$.

The ratio of $\sin{x}$ to $x$ is expressed as $\dfrac{\sin{x}}{x}$. The value of ratio of $\sin{x}$ to $x$ as $x$ approaches $0$ is written in the following mathematical form.

$\displaystyle \large \lim_{x \,\to\, 0} \normalsize \dfrac{\sin{x}}{x}$

As the angle tends to zero, a special close relation between angle and sine of angle is revealed. In this special case, the value of sine function is approximately equal to the angle. Observe the following table to understand it.

Angle $(x)$ in Radian $\sin{x}$
$0$ $0$
$0.1$ $0.099833417$ $\approx 0.1$
$0.02$ $0.019998667$ $\approx 0.02$
$0.003$ $0.002999996$ $\approx 0.03$
$0.0004$ $0.0004$
$0.00005$ $0.00005$
$0.000123$ $0.000123$
$0.0006583$ $0.0006583$
$0.00027595$ $0.00027595$
$0.000039178$ $0.000039178$

Therefore, $\sin{x} \approx x$ as $x$ approaches zero.

$\implies \displaystyle \large \lim_{x \,\to\, 0} \normalsize \dfrac{\sin{x}}{x} = \large \lim_{x \,\to\, 0} \normalsize \dfrac{x}{x}$

$\require{cancel} \implies \displaystyle \large \lim_{x \,\to\, 0} \normalsize \dfrac{\sin{x}}{x} = \large \lim_{x \,\to\, 0} \normalsize \dfrac{\cancel{x}}{\cancel{x}}$

$\implies \displaystyle \large \lim_{x \,\to\, 0} \normalsize \dfrac{\sin{x}}{x} = \large \lim_{x \,\to\, 0} \normalsize {1}$

$\,\,\, \therefore \,\,\,\,\,\, \displaystyle \large \lim_{x \,\to\, 0} \dfrac{\sin{x}}{x} = 1$

Therefore, it is proved that the value of ratio of sin function to angle is equal to one as the angle approaches zero.

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved