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# L'Hôpital's rule introduction

AP.CALC:
LIM‑4 (EU)
,
LIM‑4.A (LO)
,
LIM‑4.A.1 (EK)
,
LIM‑4.A.2 (EK)

## Video transcript

most of what we do early on when we first learn about calculus is to use limits we use limits to figure out derivatives of functions to figure out derivatives of functions in fact the definition of a derivative uses the notion of a limit it's the it's a slope around the point as we take the limit of points closer and closer to the point in question and you've seen that many many many times over in this video we're going to do I guess we're going to do it in the opposite direction we're going to use derivatives we're going to use derivatives to figure out limits to figure out limits and in particular limits that end up an indeterminate form and when I say by indeterminate form I mean that when we just take the limit as it is we end up with something like 0 over 0 or infinity over infinity or negative infinity over infinity or maybe negative infinity over negative infinity or positive infinity over negative infinity all of these are in determined indeterminate undefined forms and to do that we're going to use l'hopital's rule lop petals rule lop it hulls lop it hulls rule and in this video I'm just going to show you what lah petals rule says and how to apply it because it's fairly straightforward and it's a it's actually a very useful tool sometimes if you're in some type of a math competition they ask you to find a hard a limit that when you just plug the numbers in you get something like this lava tiles rule is normally what they are testing you for and in a future video I might approve it but that gets a little bit more involved the application is actually reasonably straightforward so what lava tiles rule tells us that if we have that if we have and I'll do it an abstract firm form first but I think when I show you the example it will all be made clear that if the limit if the limit as X approaches C of f of X is equal to 0 which is also so if this is equal to 0 and and the limit as X Roach's C of G of X is equal to zero and and this is another hand and the limit the limit as X approaches C of f prime of x over G prime of X exists and it equals L then so all of these conditions have to be met this is the inter indeterminate form of 0 over 0 so this is the first case then then we can say that the limit as X approaches C of f of X over G of X is also going to be equal to L so this might seem a little bit bizarre to you right now and I'm actually going to write the other case and then I'll do an example we'll do multiple examples of the examples we're going to make it all clear so this is the first case and the example we're going to do is actually going to be a of this case now the other case is if the limit the limit as X approaches C of f of X is equal to positive negative positive or negative infinity and and the limit as X approaches C of G of X is equal to positive or negative infinity and and the limit of I guess you could say the quotient of the derivatives exists and the limit as X approaches C of f prime of X over G prime of X is equal to L then we can make this same statement again then we can make this exact statement let me just copy that out then this again edit copy and then let me paste it so either of these two situations and then just to kind of make sure you understand what you're looking at this is the situation where if you just try to evaluate this limit right here you're going to get F of C which is 0 or the limit as X approaches C of f of X over the limit as X approaches C of G of X that's going to give you zero over zero and so you say hey I don't know what that limit is but this says well look if this limit exists I could take the derivative of each of these functions and then try to evaluate that limit and if I get a number if that exists then they're going to be the same limit this is the situation where when we take the limit we get infinity over infinity or negative infinity or positive infinity over positive or negative infinity so these are the two indeterminate forms and to make it all clear let me just show you an example because I think this will make things a lot more a lot more clear so let's say we are trying to find let's say we are trying to find the limit lose in a new color let me do it in this purplish color let's say we wanted to find the limit as X approaches zero of sine of X over X now if we just view this if we just try to evaluate it at 0 or take the limit as we approach 0 in each of these functions we're going to get something that looks like 0 over 0 sine of 0 is 0 or the limit as X approaches 0 of sine of X is 0 and obviously as X approaches 0 of X that's also going to be 0 so this is our indeterminate form and if you want to think about it this is our f of X that f of X right there is a sine of X and our G of X this G of X right there for this first case is is the X right G of X is equal to X and f of X is equal to sine of X and notice well we definitely know that this meets the first two constraints the limit as X and in this case C is 0 the limit as X approaches 0 of sine of X is 0 and the limit as X approaches 0 of X is also equal to 0 so we get our indeterminate form so let's see at least whether this limit even exists if we take the derivative of f of X and we put that over the derivative of G of X and take that the limit as X approaches 0 in this case X approaches 0 that's our C let's see if this limit exists so I'll do that in the blue so let me take so let me write the derivatives of the two functions so f prime of X if f of X is sine of X what's F prime of X well it's just cosine of X we've learned that many times and if G of X is X what is G prime of X that's super easy the derivative of X is just 1 so let's see let's try to take the limit let's try to take the limit as X approaches 0 of f prime of x over G prime of X over their derivatives so that's going to be the limit as X approaches 0 of cosine of x over over 1 and this then I wrote that one little strange over 1 and this is pretty straightforward what is this going to be well as X approaches 0 of cosine of X that's going to be equal to 1 that's equal to 1 and obviously the limit as X approaches 0 of 1 that's also going to be equal to 1 so in this situation we just saw that the limit the limit as X approaches RC in this case is 0 as X approaches 0 of f prime of X over G prime of X is equal to 1 this limit exists and it equals 1 so we've met all of the conditions this is the case we're dealing with limit as X approaches 0 of sine of X is equal to 0 limit as X approaches 0 of X is also equal to 0 the limit of the derivative of sine of X over the derivative of X which is cosine of X over 1 we found this to be equal to 1 so all of these the all of these kotappa conditions are met so then we know this must be the case that the limit as X approaches 0 of sine of X over X must be equal to 1 it must be the same it must be the same limit as this value right here where we take the derivative of the f of X and of the G of X I'll do more examples in the next few videos and I think it'll make it a lot more concrete