Math Doubts

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

Formula

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

Introduction

$x$ is a variable, $a$ and $n$ are constants. The quotient of difference of $n$-th powers of $x$ and $a$ by the difference of $x$ and $a$ is often appeared in calculus as $x$ approaches $a$. So, a special limit rule for this algebraic function is developed in calculus for evaluating the limit of these types of functions.

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

Proof

Learn how to prove the limit of ratio of difference of $n$-th powers of $x$ and $a$ to difference of $x$ and $a$ is equal to $n$ times $a$ is raised to the power of $n-1$ in calculus.



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