L'Hospital's Rule
Transcript
Welcome to video on L'Hospital's Rule. In this video, we be learning about what is L'Hospital's Rule and how can we apply this rule to solve several questions on limits. Learning this rule is important because there are several kind of questions, which can be solved fairly easily using L'Hospital's rule. And solving those questions using some other methods might be very tricky or a tedious solution would be there, and so if you use L'Hospital's rule instead, you will be saving a lot of time and energy.
And definitely that is very useful in the exam, because there is a time crunch. Okay, so before jumping to the L'Hospital's Rule let us see what are indeterminate forms. So suppose you have to evaluate limit, x tending to a, f(x), okay. If you substitute x = A directly, and if you get one of the following forms, like say, 0/0, or infinity/infinity, or infinity- infinity or say one to the power of infinity, or 0 to the power of 0, or infinity to the power of 0, or infinity into 0, then it's called an indeterminate form.
Okay, like some examples are say limit, x tending to 0 sine x/x. Okay, so this is of the form 0/0, right. So hence it is indeterminate form. Similarly, say if you have limit, x tending to 1, x squared- 1 / x minus 1.
So this is again a form 0 / 0, which is an indeterminate form, right. Now, you'll see that some of limits which are in some of these indeterminate forms, they can be solved using L'Hospital's rule. So let's see what is L'Hospital's rule. So basically, L'Hospital's rule states that if limit, x tending to a, f(x) / g(x) reduces to 0 / 0, or infinity / infinity, okay, so if the limit, x tending to a, f(x) / g(x) is of the form 0 / 0 or infinity / infinity, then, numerator and denominator until this form is reduced, okay.
So what it means is, if this is the indeterminate form that this limit is in, then, Limit, x tending to a, f(x) / g(x) = limit, x tending to a, f dash x / by g dash x, okay. So, this is what the rule says.
And, again, if limit, x tending to a, f dash x / g dash x, it reduces again to the form 0 / 0 or infinity / infinity. Then you can equate it to the limit, x tending to a, f double-dash x / g double-dash x and so on. The form 0 / 0 or infinity / infinity is removed, okay. Important thing to note here is that L'Hospital's Rule is only applicable only when the indeterminate form of the limit is 0 / 0 zero or infinity / infinity.
There is no other indeterminate form where the L'Hospital's Rule can be used, okay. So this is a very important point remember that many students sometimes tend to forget. Okay, so now we discuss what L'Hospital's Rule is. Now let's see an example where L'Hospital's Rule solves the question very easily, and there's another question which has a slightly bigger solution, so let's see that.
So suppose the question is evaluate limit x tending to 0, Root of x + a- root of a / x. Now pause this video and try to come up with a solution for this limit okay. So now let's see what will be the solution of this question if you don't use L'Hospital's Rule, okay.
So, like this thing, limit, x tending to 0 x + a root- root a / x, okay. So this is = limit, x tending to 0, root of x + a- root a / x. Into root of x + a + root a / root of x + a + root a, right.
Basically I'm just multiplying, I'm dividing this thing by the same term, okay. So now, if you just multiply it, what you get is, limit, x tending to 0, x ( root of x + a + root a) into, now this thing, this is from a + b, a- b, right, a- b and a + b.
So this becomes like a squared- b squared, right. So here it becomes like (x + a)- a, okay. So you can see that this xx cancels out and then this x will cancel out with this. So this turns into, limit, x tending to 0, 1 / root x + a + root a. Now you can substitute it to x tending to 0 here, so what you get is 1 / 2 of root a, right.
So this is one of the ways to solve this question. Now, I'll tell you another way to solve this question. So we have to evaluate limit x tending to 0, root of x + a- root a / x, right. This was to be evaluated. Now we can see that this is of the form 0 / 0, right.
So if you put x = 0, so this becomes 0, and the numerator also becomes 0. So this is of the indeterminant form, 0 / 0. So we can apply L'Hospital's rule here. So this thing is equals to limit x tending to 0. If you differentiate the numerator you get 1/2, x + a- 0, if you differentiate the denominator, you'll be getting 1, right.
Now you can directly put x = 0 here so what you'll be getting is 1 / 2 root a. Now this is the same answer that we got in the previous slide when we saw this expession using the other method. You can see that the solution in which we apply this L'Hospital's rule, it is very much shorter as compared to the previous one, right.
Sometimes it so happens, that striking or some other solution might be a bit tricky. But this L'Hospital's Rule is pretty straight forward. It just says that if you find an indeterminate form 0 / 0 and infinity / infinity, you can directly apply this L'Hospital's Rule. So, just practice more questions on this rule, and you'll be very proficient in this one.
Read full transcriptAnd definitely that is very useful in the exam, because there is a time crunch. Okay, so before jumping to the L'Hospital's Rule let us see what are indeterminate forms. So suppose you have to evaluate limit, x tending to a, f(x), okay. If you substitute x = A directly, and if you get one of the following forms, like say, 0/0, or infinity/infinity, or infinity- infinity or say one to the power of infinity, or 0 to the power of 0, or infinity to the power of 0, or infinity into 0, then it's called an indeterminate form.
Okay, like some examples are say limit, x tending to 0 sine x/x. Okay, so this is of the form 0/0, right. So hence it is indeterminate form. Similarly, say if you have limit, x tending to 1, x squared- 1 / x minus 1.
So this is again a form 0 / 0, which is an indeterminate form, right. Now, you'll see that some of limits which are in some of these indeterminate forms, they can be solved using L'Hospital's rule. So let's see what is L'Hospital's rule. So basically, L'Hospital's rule states that if limit, x tending to a, f(x) / g(x) reduces to 0 / 0, or infinity / infinity, okay, so if the limit, x tending to a, f(x) / g(x) is of the form 0 / 0 or infinity / infinity, then, numerator and denominator until this form is reduced, okay.
So what it means is, if this is the indeterminate form that this limit is in, then, Limit, x tending to a, f(x) / g(x) = limit, x tending to a, f dash x / by g dash x, okay. So, this is what the rule says.
And, again, if limit, x tending to a, f dash x / g dash x, it reduces again to the form 0 / 0 or infinity / infinity. Then you can equate it to the limit, x tending to a, f double-dash x / g double-dash x and so on. The form 0 / 0 or infinity / infinity is removed, okay. Important thing to note here is that L'Hospital's Rule is only applicable only when the indeterminate form of the limit is 0 / 0 zero or infinity / infinity.
There is no other indeterminate form where the L'Hospital's Rule can be used, okay. So this is a very important point remember that many students sometimes tend to forget. Okay, so now we discuss what L'Hospital's Rule is. Now let's see an example where L'Hospital's Rule solves the question very easily, and there's another question which has a slightly bigger solution, so let's see that.
So suppose the question is evaluate limit x tending to 0, Root of x + a- root of a / x. Now pause this video and try to come up with a solution for this limit okay. So now let's see what will be the solution of this question if you don't use L'Hospital's Rule, okay.
So, like this thing, limit, x tending to 0 x + a root- root a / x, okay. So this is = limit, x tending to 0, root of x + a- root a / x. Into root of x + a + root a / root of x + a + root a, right.
Basically I'm just multiplying, I'm dividing this thing by the same term, okay. So now, if you just multiply it, what you get is, limit, x tending to 0, x ( root of x + a + root a) into, now this thing, this is from a + b, a- b, right, a- b and a + b.
So this becomes like a squared- b squared, right. So here it becomes like (x + a)- a, okay. So you can see that this xx cancels out and then this x will cancel out with this. So this turns into, limit, x tending to 0, 1 / root x + a + root a. Now you can substitute it to x tending to 0 here, so what you get is 1 / 2 of root a, right.
So this is one of the ways to solve this question. Now, I'll tell you another way to solve this question. So we have to evaluate limit x tending to 0, root of x + a- root a / x, right. This was to be evaluated. Now we can see that this is of the form 0 / 0, right.
So if you put x = 0, so this becomes 0, and the numerator also becomes 0. So this is of the indeterminant form, 0 / 0. So we can apply L'Hospital's rule here. So this thing is equals to limit x tending to 0. If you differentiate the numerator you get 1/2, x + a- 0, if you differentiate the denominator, you'll be getting 1, right.
Now you can directly put x = 0 here so what you'll be getting is 1 / 2 root a. Now this is the same answer that we got in the previous slide when we saw this expession using the other method. You can see that the solution in which we apply this L'Hospital's rule, it is very much shorter as compared to the previous one, right.
Sometimes it so happens, that striking or some other solution might be a bit tricky. But this L'Hospital's Rule is pretty straight forward. It just says that if you find an indeterminate form 0 / 0 and infinity / infinity, you can directly apply this L'Hospital's Rule. So, just practice more questions on this rule, and you'll be very proficient in this one.
