L’Hôpital’s Rule

October 11, 2023

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Some of you may have heard of the name L’Hôpital whilst you were at school, but why was it so important? L’Hôpital’s rule, more pedantically known as “la régle de L’Hôpital”, is a highly useful technique for finding the limit of complicated expressions. To refresh your memory, the explicit definition reads: 

Where the right hand side yields an indeterminate form. This definition utilises some complex concepts. Let us break this definition down and decipher what it really means. Beginning with the limit, this is a concept that essentially forms the basis of all calculus. Limits describe how a function behaves near, but not at, a specified point. For example, if we have the function f(x) = -x2 + 4 (pictured below), and we take the limit of this function at x = 0. This means that we will look at values of the function approaching x = 0. We can do this from two sides, left and right. If we begin from the left, for example at the point x = -2, we see that f(-2) = 0 . If we take another step left, closer to x = 0, for example x = -0.5 , we have that f(-0.5)= 3.75, and another step at x = -0.2 reveals that f(-0.2) = 3.96. We can do a similar technique coming from the right-hand side, and if we continue taking infinite steps from each side, without ever taking the value x = 0, we can see where the two sides coincide: at f(x) = 4. So, we have now taken the limit of this function at x = 0, and we see that the value of this is limx-> 0 -x2 + 4 = 4.

Now, you may wonder why we use the limit if we have limx-> c f(x) = f(c), as we had in our previous example, but that is because this is not always the case. For example, if we take the same function as before, but we choose to not define our function at x = 0, i.e., we do not know what value the function takes at this point, we can still use the limit.

Here, we can see that the limit of this function is still the same, but we do not have that limx-> 0 f(x) = f(0). One final thing to note: limits from each side must approach the same value. If our left-hand side steps approach a value a , and our right-hand steps approach a value b not equal to a, then we say that the limit does not exist.

Now that we have concluded the basics of limits, we can move on to the other most important part of L’Hôpital’s rule: the indeterminate form. An indeterminant form is a difficult concept to grasp. In Layman’s terms, it is an expression that involves an operation between two functions (multiplication, division, exponents, etc.) that when the limit is taken to 0, we do not find a logical answer. This trouble with deciding what the limit should be has led to the conclusion that we are left with an indeterminate form. There are seven indeterminate forms, namely:

Looking at these, it is indeed difficult to find a limit that makes sense, so naming this an indeterminant form is quite useful. We should note that there are also cases where you may think we have an indeterminate form, but this is not the case. Specifically:

The reason in most cases being that the limit of expressions such as these are simply undefined, or have a logical answer.

Hopefully we have this concept down pat, and we can begin to use it. If we are working with limits, indeterminate forms are a tell-tale sign that further analysis is needed. It is an indication that our answer does not just stop at our indeterminate form, leaving our conclusion undefined or having no solution, but instead that we need to delve deeper.

Combining the two concepts we have just discussed, limits and indeterminate forms, we can see L’Hôpital’s rule emerge. Our limit tells us how the function f(x)/g(x) behaves as this approaches a specified value c, and when this gives an indeterminate form, we know we have to dig further in our function analysis.

So, it is clear we have to do something more, and theory tells us we use L’Hôpitals rule, but why? Well, what we are trying to find is information about our limit. We know that if our limit goes to an indeterminate form such as ∞/∞, we cannot say much about our limit, as we explained before. However, we may be able to find something about how quickly our limit goes to ∞/∞. The numerator could go really quickly to infinity and the denominator slowly. So, we should find a way to measure the slope of our functions, or more simply put: the rate of change. Those of you paying close attention that this is exactly what a derivative is. A derivative tells us the rate of change of our function. So, instead of looking at our function value at our limit, we can examine the derivative of each function at our limit, i.e., take the derivative of each function and then examine the limit. This should give us a specific limit value. We can repeat this as many times as we like until we can find a limit value.

This may be confusing, so let’s carry out an example. Let us examine limx-> 0 sin(x) / x. When we try to find this limit using conventional methods, we see that our function approaches 0/0, which is an indeterminate form. But, we do not know how quickly the numerator and denominator each approach 0, perhaps one approaches more quickly. So, let us look at the rate of change of the slope of the numerator and the denominator. Firstly, let us find the derivatives we need: f(x)=sin(x) , f'(x) = cos(x), g(x) = x, g'(x) = 1. When we fill in L’Hôpital’s rule we have that: limx-> 0 (sin(x) / x )= limx-> 0 (cos(x) / 1), which very clearly equals 0. Thus, we can see how our function behaves as x approaches 0 , or more formally, we have found our limit. This concludes our analysis.

After a rather in-depth discussion of L’Hôpital’s rule, you should be able to see why it is important and why it is so useful. If we could not use this, then we would have no way of telling how certain functions behave at specific points. L’Hôpital’s rule opens up a wealth of analysis that we can use for mathematics. It also allows us to side-step indeterminate forms, which we have discovered are not particularly nice. Now that we have learnt all this, perhaps you have gained a certain appreciation for this rule, and suddenly understand why this was so important in school.

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