Definition
L’Hôpital’s Rule is used to find the limit of an indeterminate form quickly and easily without having to use the conjugate or trig identities?
Our foundational rule for taking limits is not difficult – plug the number into the function and simplify.
But for some functions, this yields a value of zero over zero or infinity over infinity or one of the other indeterminate forms, as seen below.
This is called an indeterminate form, and we must transform our given function to evaluate the limit correctly.
Some techniques we use for evaluating limits of indeterminate forms include
- Factoring
- Common Denominators
- Conjugate Pairs
- Trig Identities
- Horizontal Asymptotes For Limits At Infinity
But sometimes, even these methods can be challenging to implement so we can use L’Hôpital’s Rule for evaluating indeterminate limits! 1
Who
Continuous Compounding Interest Rates
L’Hospital’s Rule is used to prove that the compound interest rate equation through continuous compounding equals Pe^rt. (Manacheril)
Continuous compounding interest rates encountered everyday in investments, different types of bank accounts, or when paying credit cards bills, mortgages, etc. 3
The Gamma Function
The Gamma function is used to model the factorial function. Because the common way to determine the value of n! was inefficient for large n‘s, the gamma function was created, an integral formula for n!. L’Hopital’s rule is used in proving the Gamma Function along with using integration by parts. (Gamma Function)
The Gamma Function is used in engineering, quantum physics, astrophysics, fluid dynamics, statistics, combinatorics, probability theory, etc. Specifically, it is often used in Gamma distributions, which are used in statistical models for predicting different kinds of natural events including the interval of time between earthquakes. (Rice) 3
What
We can apply L’Hôpital’s Rule whenever direct substitution of a limit yields an indeterminate form. 1
The L’Hôpital’s rule is often misused. The indeterminate forms for the L’Hôpital’s rule to apply are 0/0, 0×∞, ∞/∞, ∞ − ∞, ∞⁰, 0⁰, and 1^∞. We often forget about the indeterminate forms, for example, ∞/0 — though still undefined– is not an indeterminate form for the L’Hôpital’s rule. When finding limit of lnx/(1/x) ,as x approaches infinity, we get ∞/0. We can’t apply the L’Hôpital’s rule here. We will get the limit 0 but the limit is ∞. 5
Why
In the US, outside of math majors (and parts of High School Geometry), math is taught not for its own sake, but for its application to science and engineering. So calculus is taught with its original intent – as a tool for calculation. L’Hopital’s rule is a workhorse for calculating limits, which is why it’s taught early (usually immediately after differentiation). 4
See Theoretical Knowledge Vs Practical Application.
How
Why Does L’Hôpital’s Rule Work
Suppose we are taking the limit of a function as x approaches infinity and whose numerator and denominator subsequently approach infinity as well.
This means the limit of this algebraic fraction is indeterminate, and the limit doesn’t give us a clear picture of what is happening.
Is the numerator rapidly approaching infinity while the denominator is going to infinity more slowly? Or is the denominator speeding toward infinity, and the numerator is lagging behind? In other words, who is dominating the behavior of the overall limit? Is it the numerator or denominator?
Is there a way to observe their individual rate of change, then we can see which function is the dominant function.
Rather than just comparing the numerator and denominator directly, we compare their derivatives (i.e., rate of change)!
And this is the idea behind L’Hôpital’s Rule!
By comparing the rate of change, we can easily decipher the function’s behavior as a whole. 1
How To Use L’Hôpital’s Rule
We differentiate the numerator and the denominator separately and then take the limit.
Additionally, I would like to point out that there will be times when L’Hopital’s Rule must be applied more than once to calculate the limit value successfully. 1
Example #1
Let’s look at how we apply L’Hopital’s Rule for the indeterminate form of zero over zero.
Notice that we could have just as easily factored the rational function and arrived at the same answer, but with L’Hôpital’s Rule, we achieved the same goal using derivatives. 1
If possible, I suggest you graph the function to see how it behaves.
L’Hôpital’s Rule – Indeterminate – Divided By Zero using GeoGebra
Example #2
Let’s look at how we can use L’Hôpital’s Rule rule for the indeterminate form of infinity over infinity.
Observe that we had to apply L’Hôpital’s Rule twice to find the limit value.
But overall, the process is straightforward: if the limit is indeterminate, take the derivative of the top and the derivative of the bottom separately and then reevaluate the limit until you arrive at a defined value. 1
If possible, I suggest you graph the function to see how it behaves.
Important Notes on L’Hospital’s Rule:
- The limit of a fraction of two functions (that results in an indeterminate form) is equal to the limit of the fraction of their derivatives.
- Do not apply L’Hopital’s rule if the limit is not resulting in an indeterminate form.
- We can apply L’Hopital’s rule as many times as required but before the application of each time, we should check whether the limit in that particular step is giving indeterminate form.
- When we are trying to apply L’Hopital’s rule for the product f(x) · g(x), first, write it as fraction (i.e., either as f(x) / (1 / g(x)) or as g(x) / (1/ f(x)) ). 2
Many of the References and Additional Reading websites, and Videos will assist you with learning and applying L’Hôpital’s Rule.
As some professors say: “It is intuitively obvious to even the most casual observer.“
References
1 “L’Hopital’s Rule”. 2022. CalcWorkshop. https://calcworkshop.com/derivatives/lhopitals-rule/.
2 “L’hopital’s Rule – Formula, Proof | L’hospital’s Rule”. 2022. CUEMATH. https://www.cuemath.com/calculus/l-hopitals-rule/.
3 “Real World Applications”. 2022. L’Hospital’s Rule. https://lhospitalsrule.weebly.com/real-world-applications.html.
4 “Why is L’Hôpital’s Rule so prominent in the American calculus curriculum?” 2022. Quora. https://qr.ae/pvowWe.
5 Hamxa, Muhammad. “The Trap of L’Hôpital’s Rule”. 2022. Medium. https://medium.com/@hamxa26/the-trap-of-lh%C3%B4pital-s-rule-94145e40fe7e.
Additional Reading
” Calculus I – L’Hospital’s Rule And Indeterminate Forms “. 2022. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/Classes/CalcI/LHospitalsRule.aspx.
“L’Hôpital’s Rule”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/l-hopitals-rule.html.
“L’Hôpital’s rule – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule.
“L’Hopital’S Rule – Calculus Tutorials”. 2022. math.hmc.edu. https://math.hmc.edu/calculus/hmc-mathematics-calculus-online-tutorials/single-variable-calculus/lhopitals-rule/.
“L’Hôpital’s Rule Introduction (Video) | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-contextual-applications-new/ab-4-7/v/introduction-to-l-hopital-s-rule.
“What is an intuitive explanation of l’Hopital’s Rule?” 2022. Quora. https://qr.ae/pvowHX.
Videos
This calculus video tutorial provides a basic introduction into L’Hôpital’s Rule. It explains how to use L’Hôpital’s Rule to evaluate limits with trig functions, fractions, exponential functions with ex and natural log functions such as ln(x). To use L’Hôpital’s Rule, you need to take the derivative of the numerator and denominator of the fraction any time you have an indeterminate form as a result of direct substitution. This calculus video contains plenty of examples and practice problems on L’Hôpital’s Rule rule.
This calculus video tutorial explains the concept of L’Hôpital’s Rule and how to use it to evaluate limits associated with indeterminate forms of zero and infinity. This video contains plenty of examples with natural logs, trig functions, and exponential functions. It contains plenty of practice problems for you to work on.
Mathematical Mysteries 