Introduction to L'Hôpital's Rule Introduction to L'Hôpital's Rule
Introduction to L'Hôpital's Rule
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- 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.
- In fact, the definition of a derivative uses
- the notion of a limit.
- 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 I guess we're going to do it in the
- opposite direction.
- We're going to use derivatives to figure out limits.
- And in particular, limits that end up in 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 0/0, 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 indeterminate, undefined forms.
- And to do that we're going to use l'Hopital's rule.
- And in this video I'm just going to show you what
- l'Hoptial's rule says and how to apply it because it's fairly
- straightforward, and it's actually a very useful tool
- sometimes if you're in some type of a math competition and
- they ask you to find a difficult limit that when you
- just plug the numbers in you get something like this.
- L'Hopital's rule is normally what they are testing you for.
- And in a future video I might prove it, but that gets a
- little bit more involved.
- The application is actually reasonably straightforward.
- So what l'Hopital's rule tells us that if we have-- and I'll
- do it in abstract form first, but I think when I show you
- the example it will all be made clear.
- That if the limit as x roaches c of f of x is equal to 0, and
- the limit as x approaches c of g of x is equal to 0, and-- and
- this is another and-- and 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 indeterminate form of 0/0, so this
- is the first case.
- 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 and the examples are 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 an example of this case.
- Now the other case is if the limit as x approaches c of f of
- x is equal to positive or negative infinity, and the
- limit as x approaches c of g of x is equal to positive or
- negative infinity, 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.
- Let me just copy that out.
- Edit, copy, and then let me paste it.
- So in either of these two situations just to kind of make
- sure you understand what you're looking at, this is the
- situation where if you just tried 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 0/0.
- 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 a 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 clear.
- So let's say we are trying to find the limit-- I'll
- do this in a new color.
- Let me do it in this purplish color.
- Let's say we wanted to find the limit as x approaches
- 0 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 0/0.
- 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 the sine of x.
- And our g of x, this g of x right there for this
- first case, is the x.
- 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 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 the limit as x approaches 0
- in this case, that's our c.
- Let's see if this limit exists.
- So I'll do that in the blue.
- 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.
- You'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.
- 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 1.
- I wrote that 1 a little strange.
- 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.
- 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 as x
- approaches-- our c 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 z 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.
- All of these top 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 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.
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