Math Doubts

Power difference rule of Limits in Ratio form

Formula

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{x^{\displaystyle n}-a^{\displaystyle n}}{x-a} \,=\, n.a^{\displaystyle n-1}}$

Introduction

Let $x$ be a variable, and $a$ and $n$ be two constants. Let’s assume that two quantities are expressed in exponential form as $x^{\displaystyle n}$ and $a^{\displaystyle n}$.

  1. The difference of the quantities in exponential notation is written as $x^{\displaystyle n}-a^{\displaystyle n}$.
  2. The subtraction of the constant $a$ from the variable $x$ is written as $x-a$.

The ratio of the above two indeterminate quantities is written as follows in mathematics.

$\dfrac{x^{\displaystyle n}-a^{\displaystyle n}}{x-a}$

The limit of this rational expression as the value of $x$ approaches to $a$ is written in the following mathematical form.

$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{x^{\displaystyle n}-a^{\displaystyle n}}{x-a}}$

The limit of the $x$ raised to the power $n$ minus $a$ raised to the power $n$ by $x$ minus $a$ as the value of $x$ is closer to $a$, is equal to the $n$ times $a$ raised to the power $n$ minus $1$.

$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{x^{\displaystyle n}-a^{\displaystyle n}}{x-a}}$ $\,=\,$ $n \times a^{\displaystyle n-1}$

It can be called the power-difference limit rule in ratio form.

Proof

Learn how to derive the power-difference law of limits in ratio form in calculus mathematically.

Problems

List of the questions on power-difference property of limits in ratio form with solutions.

Math Doubts

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.

Copyright © 2012 - 2021 Math Doubts, All Rights Reserved