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# L’Hôpital’s rule (composite exponential functions)

## Video transcript

what I would like to tackle in this video is what I consider to be a particularly interesting limits problem so let's say we wanted to figure out the limit as X approaches zero from the positive direction of sine of X and this is where it's about to get interesting sine of X to the 1 over the natural log of X power and I encourage you to pause this video and see if you can have a go at it fully knowing that this is a little bit of a tricky exercise so I'm assuming you have attempted some of you might have been able to figure it out in a first pass I will tell you that the first time that I encountered something like this I did not figure it out I first vet so definitely do not feel bad if you fall into that second category because what money of you all probably did is you said okay let me think about it let me just think about the components here if I were to think about the limit if I were to think about the limit as X approaches zero from the positive direction of sine of X well that's pretty straightforward that's going to be 0 so you could think of like this part of it is going to approach zero but then if you say and and you could say I guess I should say the limit as X approaches zero from the positive direction of 1 over natural log of X and this is why we have to think about it from the positive direction it doesn't make sense to approach it from the negative direction you can't take the natural log of a negative number that's not in the domain for the natural log but as you get closer and closer to 0 from the negative direction the natural log of those values you have to raise e to more larger and larger negative values this part over here is going to approach negative infinity he's going to go to negative infinity 1 over negative infinity 1 divided by super large or large magnitude negative numbers well that's just going to approach 0 so you could say that this right over here is also going to be is also going to be equal to 0 but that doesn't seem to help us much because if this thing is going to 0 and that thing is going to 0 it's kind of an implication that well maybe this whole thing is going to 0 to the 0 power but we don't really know what 0 let me do this home those colors 0 to the to the zero power but this is one of those great fun things to think about in mathematics there's justifications why this could be zero justifications why this could be one we don't really know what to make of this this isn't really a a satisfying answer so something at this point might be going into your brain you had this that we have this thing that we've been exposed to called la putana rule if you have not been exposed to it I encourage you to watch the video the introductory video on lapa tiles rule and lapa tiles rule let me just write it down lapa tiles lapa tiles rule helps us out with situations where when we try to superficially evaluate the limit we get indeterminate forms things like zero over zero we get infinity over infinity we get negative infinity over negative infinity and we go into much more detail into that video and it seems like well this is kind of it feels like you know we're getting a zero to zero it's kind of we're getting the strange beast and it at least evokes the notion of lapa tiles rule and so you will not be as we'll see in a few seconds you're not wrong - for that lapa tiles rule neuron to be trigging in your brain although you can't apply it directly to this right over here lapa tiles rule does not apply to or directly apply to the zero to the zero form but what we can do is construct a problem where la patate rule will apply and then use that to solve to figure out what this is going to be and this is this was essentially the tricky part of this exercise what do I mean well if we set y equal to Y of and I maybe let me write it this way if we set and I'll write Y I could just write Y but I'll say Y is clearly a function of X if we say Y of X is going to be sine of X sine of X to the 1 over natural log of X this thing right over here is essentially saying what's the limit as X approaches zero from the positive direction of Y and once again we don't know it's going to maybe it's a zero to the zero but we don't know what zero to the zero actually is but what we could do what we could do and this is a trick that you see a lot and anytime you get kind of weird things with exponents and you know whether you're doing limits or derivatives and as you'll see it's oftentimes useful to take the natural log of boats well what happens if you take the natural log of both sides here so on the left hand side you're going to have the natural log the natural log whenever I think of natural log and the no and either way I was think of the the color green for some bizarre reason but we'll say the natural log of Y is equal to if you take the natural log of this thing actually let me just I don't want to skip steps here because this isn't interesting so it's going to be the natural log of all of this business of sine of X right this way sine of X sine I want to do this in that orange color the natural log of sine of X to the 1 over the natural log of X well we know from our exponent power our logarithm properties then at the logarithm of something to a power that's the same thing as the power 1 over natural log of x times the logarithm in this case the natural logarithm of whatever we're taking the of sine of X here sine of sine of X or we could say the natural log of Y which I want to keep state color consistent for at least one more step the natural log of Y is equal to if we just rewrite this this is going to be the natural log of sine of X the natural log of sine of X over the natural log of X well this is all interesting but why do we care about this why did I do this well instead of thinking about what is the limit what is the limit as X approaches zero from the positive direction of Y let's think about what the natural log of Y is approaching as we approach as X approaches zero from the positive direction so let's figure out what the limit of this expression right over here is as X approaches zero from the positive direction what is the natural log of Y what is this whole thing not Y what is the natural log of Y approaching so let's think about that scenario so let me write do this in a new color so we want to figure out what is the limit as X approaches zero from the positive direction of this business and I'll just write in one color the natural log of sine of X over the natural log of X and I wrote one time in print and one time in cursive I'll just be consistent right over there now why is this interesting well let's see in the numerator here this thing is going to approach is zero natural log of zero you're going to proach a negative infinity this thing right over here natural log of as you approach from the positive direction once again you're going to approach negative infinity this gives you that indeterminate form so this is this is giving you that indeterminate form of negative infinity over negative infinity which is neat because this triggers or at least tells us that lapa tiles rule may be appropriate here so we could say that this is going to be equal to the limit as X approaches zero from the positive direction let me give myself a little bit more real estate to work up with and I can take the derivative of the numerator and the derivative of the denominator derivative of the numerator so the derivative of the numerator ply the chain rule here derivative of sine of X is cosine of X and then and then the derivative of the natural log of sine of X with respect to sine of X is going to be 1 over sine of X so it's just going to be over sine of X so that's the derivative of the numerator and then the derivative of the denominator is just going to be it's just going to be 1 over X 1 over X so this is all going to be equal to this is equal to the limit as X approaches 0 from the positive direction of I could write this as cosine of X cosine of X and let's see if I'm dividing if I'm dividing by X so I'm dividing by X I am going to get so this is going to be x over sine of X X over sine of X and when I apply and when I try to take the limit here I'm going to zero once again we get a 0 over 0 so this doesn't feel too satisfying but once again this is where our limit property might be useful and this as you can tell this is not the most trivial of problems but this is going to be the same thing and this took a look this would take a little bit of pattern recognition this is the same thing as because we know that the limit of the product of two functions is equal to the product of their limits this is the same thing as the limit the limit as X approaches zero from the positive direction of if we take this part let me do this in a different color if we take this point that's not a different color if we take this part right over here so that's going to be x over sine of X and then times the limit so I put parentheses here times the limit the limit as X approaches zero from the positive direction of cosine of X of cosine of X now this thing right over here is pretty straightforward you can just evaluate it at 0 you get 1 so this thing right over here is equal to 1 but what's this thing and this might ring a bell you might have seen the limit as X approaches 0 sine of X over X this is just the reciprocal of that this is x over sine of X but when you just superficially try to evaluate it you get 0 over 0 so you can then apply l'hopital's rule to this thing so once again this is quite quite an interesting scenario we find ourselves in so this is going to be the same thing as the limit as X approaches 0 from the positive direction derivative of the top is 1 derivative the bottom is cosine of X well this is just going to be 1 over cosine of 0 is 1 so this is just going to be equal to 1 so we have to apply l'hopital's rule again and realize that this limit is going to be equal to 1 so 1 times 1 is 1 so this thing right over here is equal to 1 so this thing right over here this thing right over here is going to be this thing right over here is going to approach 1 which tells you that this thing is approaching 1 so what do we now know we now know and I'll write it out in language we now know that the limit the limit of the natural log of Y the limit of the natural log of Y as X approaches zero from the positive direction is equal to one so if the if the natural log of Y is approaching one what is y approaching well in order for the natural log so once again we just know this thing right over here we know this thing is 1 and this thing is the natural log of Y we now know we now know that this thing the limit as X approaches zero of this thing is 1 and that's the same thing as the limit as X approaches 0 from the positive direction of the natural log of Y these things are equivalent is equal to 1 well if the natural log of Y is approaching 1 so if the natural log of Y is approaching let me write it this way if the natural log of Y is approaching 1 well what must Y be approaching well to get the natural log of something that gets you 1 well y must be approaching e because natural log of e is 1 so then Y must be approaching e and we are done because that's what we cared about we cared about what is y this is why I remember we defined this whole thing as Y we said what is Y approaching as X approaches zero from the positive direction well we figure out that the natural log of Y is approaching 1 as X approaches zero from the positive direction so that means that y must be approaching e so this is this is tells us that this thing this thing right over here is equal to e which is somewhat mind-blowing we see another you know the e is popping up and it's involving sine of X and of course natural log of X you expect e to be involved somehow but it is a pretty fascinating problem in my mind