Math Doubts

$\displaystyle \lim_{x \to 0} \dfrac{\sin{x}}{x}$ formula


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

The limit of quotient of sin of an angle by angle as the angle approaches zero is equal to one. It is a most useful limit rule for dealing trigonometric functions especially sine in calculus.


$x$ is a variable and also represents angle of the right triangle. The limit of quotient of sin of angle $x$ by angle $x$ as the angle $x$ tends to zero is written mathematically in calculus as follows.

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

The trigonometric limit property can be derived in mathematics in two different methods.

Relation of Sine function and angle

It is derived on the basis of close relation between $\sin{x}$ function and angle $x$ as the angle $x$ closer to zero.

Taylor (or) Maclaurin Series Method

It can also be derived by the expansion of sinx function as per Taylor (or) Maclaurin series.

Follow us
Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more