LHopitalsRule

L'Hôpital's Rule is a mathematical theorem used to find the limit of indeterminate forms, particularly of the type $0/0$ or $\infty/\infty$. It's named after the French mathematician Guillaume de l'Hôpital, who published the rule in his book "Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes" in 1696, although the rule was actually discovered by the Swiss mathematician Johann Bernoulli.

Definition:

L'Hôpital's Rule states that if the functions $f(x)$ and $g(x)$ are differentiable and $g'(x)$ is not zero near a point $c$, and if

$$ \lim_{{x \to c}} f(x) = 0 $$

and

$$ \lim_{{x \to c}} g(x) = 0 $$

(the $0/0$ case)

or

$$ \lim_{{x \to c}} f(x) = \pm\infty $$

and

$$ \lim_{{x \to c}} g(x) = \pm\infty $$

(the $\infty/\infty$ case),

then the limit of their quotient as $x$ approaches $c$ can be found by taking the limit of the quotients of their derivatives:

$$ \lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)} $$

provided that the limit on the right side exists or is ±∞.

Conditions and Usage:

1. Differentiability: Both $f(x)$ and $g(x)$ must be differentiable in an interval around $c$ (except possibly at $c$ itself).

2. Indeterminate Form: The rule applies when the limit yields an indeterminate form of $0/0$ or $\infty/\infty$.

3. Existence of Limit: The limit of $f'(x)/g'(x)$ must exist or be infinite.

4. Repeated Application: If the resulting limit after applying L'Hôpital's Rule is still an indeterminate form, the rule can be applied repeatedly.

Examples:

1. Simple Case:

$$ \lim_{{x \to 0}} \frac{\sin(x)}{x} $$

Both numerator and denominator approach 0 as $x$ approaches 0. Applying L'Hôpital's Rule:

$$ \lim_{{x \to 0}} \frac{\cos(x)}{1} = 1 $$

2. Repeated Application:

$$ \lim_{{x \to 0}} \frac{e^x - 1 - x}{x^2} $$

Initially, this gives $0/0$. Applying L'Hôpital's Rule once:

$$ \lim_{{x \to 0}} \frac{e^x - 1}{2x} $$

Still $0/0$, apply the rule again:

$$ \lim_{{x \to 0}} \frac{e^x}{2} = \frac{1}{2} $$

Limitations:

• It doesn't apply to forms other than $0/0$ or $\infty/\infty$.
• The existence of the derivatives of $f$ and $g$ is crucial.
• The existence of the limit of $f'(x)/g'(x)$ is necessary.

Conclusion:

L'Hôpital's Rule is a powerful tool in calculus for evaluating limits of indeterminate forms, but it's important to apply it under the correct conditions and with an understanding of its limitations.