# Power difference rule of Limits in Ratio form

## Formula

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

### 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 $x^n$ and $a^n$.

1. The difference of the quantities in exponential notation is written as $x^n}-a^n$.
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.

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

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

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

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 $1$.

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

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.

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