L’Hospital’s Rule: Solving Indeterminate Forms
by Yuvi K - January 1, 2024
What is L’Hospital’s Rule?
L’Hospital’s Rule is an invaluable tool for solving a type of mathematical equation known as an Indeterminate Form. An Indeterminate Form is a fraction whose denominator and numerator both approach infinity at the same rate; these equations have no defined answer. Fortunately, L’Hospital’s Rule offers a solution.
To better visualize L’Hospital’s Rule, we can compare it to an elevator. If two passengers—one on the first floor and one on the ninth floor—want to meet, they must both get to the same floor. If an elevator is not functioning, they could try using the stairs; however, this is a slow and tedious process that could take hours or even days. With an elevator, they could simply arrive at their destination quickly and efficiently. L’Hospital’s Rule is like this elevator; it enables us to find the answer to Indeterminate Forms quickly and accurately.
History of L’Hospital’s Rule:
L’Hospital’s Rule was first derived in 1696 by Marquis de L’Hospital, a renowned French mathematician, physicist, and astronomer. The Marquis was working to solve a certain type of mathematical equation comprised of fractions with no defined answer when he discovered the Rule. His work was eventually published in his book, Analyse des Infiniment Petits pour l’intelligence des lignes courbes, making L’Hospital’s Rule one of the first established properties of calculus.
How L’Hospital’s Rule Works:
L’Hospital’s Rule is used to solve Indeterminate Forms—a type of mathematical equation composed of fractions with no defined solution. To utilize the Rule, one must first identify if an equation is an Indeterminate Form. Then, the Rule is used to take the derivatives of each side of the equation and to simplify it into a regular fraction with a defined answer.
Using L’Hospital’s Rule Step-by-Step:
Using L’Hospital’s Rule is not a complex process; here is a step-by-step guide for doing so.
Step 1: Identify the Indeterminate Forms
The first step is to identify if the equation is an Indeterminate Form. An example of this is
(limx→a) [f(x)/g(x)] = ?
Step 2: Apply L’Hospital’s Rule
Once an Indeterminate Form is spotted, the Rule can be applied and used to take the derivatives of each side of the equation and to simplify it into a regular fraction with a defined answer.
For example:
f'(x)/g'(x) = limx→a [f(x)/g(x)]
Step 3: Solve the Equation
The next step is to solve for the equation. To do this, derivatives of both sides must be taken, resulting in a fraction with a defined solution.
For example:
f'(x)/g'(x) = (2x+1)/(x^2-1)
Conclusion:
L’Hospital’s Rule is an invaluable tool for solving Indeterminate Forms, a type of equation with no defined answer. The Rule was established in 1696 by French mathematician, physicist, and astronomer Marquis de L’Hospital. L’Hospital’s Rule is used to take the derivatives of each side of the equation and to simplify it into a regular fraction with a defined answer. Its systematic three step process makes it easy to use and understand.
