# $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin^{-1}{x}}{x}}$ formula

## Formula

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

The limit of quotient of inverse sine function by a variable as the input approaches zero is equal to one. It is a standard result in calculus and used as a formula in mathematics.

### Introduction

Assume $x$ is a variable and represents the ratio of lengths of opposite side to hypotenuse in a right triangle. The inverse sine function in terms of $x$ is written as $\sin^{-1}{x}$ or $\arcsin{x}$ in inverse trigonometry.

The limit of the $\arcsin{x}$ by $x$ as $x$ approaches zero is written in mathematical form as follows.

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

In fact, the limit of $\arcsin{(x)}/x$ as $x$ tends to $0$ is equal to $1$. It is used often appeared in calculus. So, this standard inverse trigonometric function result is used as a formula in calculus.

#### Other forms

The limit rule of inverse trigonometric function can be written in several ways in calculus.

$(1) \,\,\,$ $\displaystyle \large \lim_{m \,\to\, 0}{\normalsize \dfrac{\sin^{-1}{m}}{m}}$ $\,=\,$ $1$

$(2) \,\,\,$ $\displaystyle \large \lim_{p \,\to\, 0}{\normalsize \dfrac{\sin^{-1}{p}}{p}}$ $\,=\,$ $1$

$(3) \,\,\,$ $\displaystyle \large \lim_{y \,\to\, 0}{\normalsize \dfrac{\sin^{-1}{y}}{y}}$ $\,=\,$ $1$

### Proof

Learn how to prove that the limit of $\arcsin{(x)}/x$ as $x$ tends to zero is equal to one in calculus.

Latest Math Topics
Apr 18, 2022
Apr 14, 2022

A best free mathematics education website for students, teachers and researchers.

###### Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

###### Maths Problems

Learn how to solve the maths problems in different methods with understandable steps.

Learn solutions

###### Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.