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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.

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