Math Doubts

Power difference rule of Limits in Ratio form


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


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.


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


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

Math Doubts
Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects. Know more
Follow us on Social Media
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