L’Hôpital’s Rule or L’Hospital’s Rule

Formula

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

Introduction

The limits of the rational functions have to evaluate in calculus and their limits can be indeterminate in some cases as the value of input approaches a value. It is complicated to find the limits of them. So, it requires a special limit rule to find the limits of such rational functions.

Let $f(x)$ and $g(x)$ be two functions in terms of $x$ and assume that the limit of rational function $f(x)$ divided by $g(x)$ as the value of $x$ tends to zero is indeterminate as follows.

$(1).\,\,$ $\displaystyle \large \lim_{x\,\to\,c}{\normalsize \dfrac{f(x)}{g(x)}}$ $\,=\,$ $\dfrac{f(c)}{g(c)}$ $\,=\,$ $\dfrac{0}{0}$

$(2).\,\,$ $\displaystyle \large \lim_{x\,\to\,c}{\normalsize \dfrac{f(x)}{g(x)}}$ $\,=\,$ $\dfrac{f(c)}{g(c)}$ $\,=\,$ $\dfrac{\infty}{\infty}$

A French mathematician Guillaume de L’Hôpital introduced an advanced mathematical approach to evaluate the limit of a rational function when its limit is indeterminate as its input approaches a value. Hence, it is called the L’Hôpital’s rule and also popularly called the L’Hospital’s rule. Alternatively, it is called the Bernoulli’s rule.

Guillaume de l’Hôpital discovered that the limit of a rational function can be calculated after differentiating both expressions in the rational function when the limit of a rational function is indeterminate.

$\implies$ $\displaystyle \large \lim_{x\,\to\,c}{\normalsize \dfrac{f(x)}{g(x)}}$ $\,=\,$ $\displaystyle \large \lim_{x\,\to\,c}{\normalsize \dfrac{\dfrac{d}{dx}f(x)}{\dfrac{d}{dx}g(x)}}$

It can be written in the following form by representing the derivatives of the functions by the $f'(x)$ and $g'(x)$ respectively.

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

The differentiation of the functions in both numerator and denominator of a rational function can be continued until the limit is evaluated.

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

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

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vdots$