L’Hopital’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 limit of a rational expression is equal to indeterminate as the input of function approaches some value. In this case, it is not possible to evaluate the limit of such functions. However, a French mathematician Guillaume de l’Hôpital introduced a mathematical approach for evaluating the limit of a function whose limit equals to indeterminate as its input approaches a value. Thus, the mathematical approach is called as L’Hopital’s rule or L’Hospital’s rule.

In this method, the limit of a rational expression is calculated by evaluating the limit by the direction substitution after differentiating the expressions in both numerator and denominator until we obtain the determinate form.

Let $f(x)$ and $g(x)$ be two real functions in $x$ and $c$ is a constant. If the functions $f(x)$ and $g(x)$ form a rational expression as $\dfrac{f(x)}{g(x)}$, then the limit the rational expression as $x$ approaches $c$ is written in mathematical form as follows.

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

There are two possibilities for the limit of a function to become indeterminate. Now, let’s discuss both the cases for studying the L’Hopital’s rule in calculus.

If $f(c) = 0$ and $g(c) = 0$

If $f(c) \,=\, 0$ and $g(c) \,=\, 0$, the limit of the function is indeterminate as $x$ approaches $c$.

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

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

If $f(c) = \infty$ and $g(c) = \infty$

When $f(c) \,=\, \infty$ and $g(c) \,=\, \infty$, the limit of the function is also indeterminate as $x$ approaches $c$.

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

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

The above two cases reveal that the limit of a function can be indeterminate, which can be in either $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ form. In these cases, it is not possible to evaluate the limit of a function but the L’Hopital’s or L’Hospital’s rule makes it possible.

Guillaume de l’Hôpital had understood that the limit of a function whose value is indeterminate, is equal to the limit of the rational expression when the expressions in both numerator and denominator are differentiated until we get the determinate form by substitution.

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

In the differential calculus, $\dfrac{d}{dx}f(x)$ and $\dfrac{d}{dx}g(x)$ are simply represented by $f'(x)$ and $g'(x)$ respectively.

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

If we still get the indeterminate form even after differentiating both expressions in numerator and denominator, then repeat the differentiation process until the indeterminate form is removed.

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

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

and so on.

Proof

Learn how to prove the l’hopital’s rule fundamentally in calculus.

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