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

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.

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.

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

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

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

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

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

Mar 21, 2023

Feb 25, 2023

Feb 17, 2023

Feb 10, 2023

Jan 15, 2023

Latest Math Problems

Mar 03, 2023

Mar 01, 2023

Feb 27, 2023

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

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

Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved