Quotient of Limits Rule

Formula

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{f(x)}{g(x)}}$ $\,=\,$ $\dfrac{\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}}{\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}$

Introduction

In mathematics, an expression can be formed by division of two functions and their quotient may interrupt the procedure of finding the limit. So, a special limit formula is essential for us to evaluate the limit of a quotient. It is called the division of limits rule and it is also called the quotient of limits rule.

Let’s consider two functions and denote them as $f(x)$ and $g(x)$, whereas let’s assume that the function $f(x)$ is divided by another function $g(x)$, and their division is written as $f(x) \div g(x)$ mathematically. The limit of the division of $f(x)$ divided by $g(x)$ as the value of variable $x$ approaches to $a$ is written as follows.

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \big[f{(x)} \div g{(x)}\big]}$

Now, the limit of quotient of $f(x)$ divided by $g(x)$ as $x$ tends to $a$ can also be written in mathematics as follows.

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{f{(x)}}{g{(x)}}}$

According to quotient law of limits in calculus, the limit of a quotient is equal to quotient of their limits. So, the limit of quotient of $f(x)$ by $g(x)$ is equal to quotient of the limits of $f(x)$ by $g(x)$ as $x$ tends to $a$.

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{f{(x)}}{g{(x)}}}$ $\,=\,$ $\dfrac{\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}}{\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}$

The above equation mathematically expresses the division of limits theorem and it can be used as a formula in calculus.

Examples

Let’s understand the limit quotient rule from an easy limit problem.

Now, let’s learn more about the limit of a quotient law in calculus.