Applications of Derivatives. L'Hopital's Rule (Part I)

Applications of Derivatives. L'Hopital's Rule (Part I)

L'Hopital's rule is a method for finding limits of indeterminate forms by treating them as fractional derivatives. It is a method to evaluate the limit of a function when it is equal to some limit. The rule tells us whether the function can be differentiated or integrated, and if so, by how much.

It’s time to learn how to find limits by using L'Hopital's Rule. This rule was discovered by a French mathematician named Guillaume Leblanc--or L'Hopital--in the early 1700s.

The rule states that if you have an infinite limit in one direction and an indeterminate form in the other direction, then you can use L'Hopital's Rule to evaluate the limit by rewriting it as zero divided by zero.

Here is a formula:

Suppose we have two functions, f and g, that are differentiable on the open interval containing a. And suppose further that g is not equal to zero when x is not equal to a.

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Suppose finally that f(a) = g(a) = 0, then the limit of this ratio f(x)/g(x). When x tends to a, it's given by the limit of f'(x)/g'(x), provided that this limit exists. Why is this useful? This can be useful in various situations involving limits.

Imagine you need to compute the limit of this ratio, fx/gx when x tends to a. Very naturally, we try to put in x equals to a in the ratio here. For those difficult cases, we always find that f(a) equals to zero.

However, since both f(a) and g(a) equal zero, this ratio becomes meaningless, and we cannot proceed further using traditional methods. Now the L'Hopital's Rule is applicable exactly in this difficult case.

If you have zero over zero, you can convert the limit of this ratio to this ratio with f and g swapped. This new ratio has a limit when x tends to a and that limit is actually equal to the limit of the original fraction f over g.

Thus, the condition for applying L'Hopital's Rule is that the problem be difficult.

Mainly, f(a)=0 and g(a)=0. Secondly, this new fraction has a limit when x tends to a. Under these conditions, the L'Hopital's Rule can be applied. The proof of this theorem depends on the following theorem called Cauchy's Mean Value Theorem.

We tried to evaluate the limit as x approaches 0 of this fraction. We have to check if L'Hopital's Rule applies here. If we set x equal to 0, we have 1-x=-1 and x=0. So both numerator and denominator are equal to zero--this satisfies the condition for application of

L'Hopital's Rule. According to L'Hopital's Rule, you can convert the limit of this expression by differentiating the top and bottom separately.

Now, be careful. We differentiate the top and get sine x. We differentiate the bottom and get 2x. We differentiate these separately - we do not apply the Portion Rule to them. This is not a Portion Rule problem! You just differentiate the top and bottom separately.

Therefore, according to L'Hopital's Rule, the limit of this original expression is now converted to the limit of this new expression. If we put x equals zero, we get zero in both top and bottom. So again we have a situation in which L'Hopital's Rule can be applied again here.

You differentiate the top and bottom, and you get cosine x and 2. So we want to find the limit of this expression when x tends to zero. You try putting zero in the top; you get one. You try putting zero here; you get 2.

So in this fraction you do not get zero and zero. So you cannot apply the L'Hopital's Rule again here. You have to stop here. Now put x equals zero, you get one over two as the answer. So the answer to this problem is one over two.

So you remember that when we apply L'Hopital's Rule, we must check first both the numerator and denominator equal to zero when x = a. The form we saw in the example above is indeterminate because it contains zero over zero.

There are other indeterminate forms as well, like the following: infinity over infinity. If you put in the value a into the fraction, both the top and bottom are equal to infinity. For some expressions, it is of the form, infinity times zero. Or in some forms, infinity minus infinity or one infinity to the power of zero.

So all of these are called indeterminate forms. Basically, you can understand that these are the functions we cannot find the limit of by simply plugging in numbers. We have to use L'Hopital's Rule in this situation. L'Hopital's Rule is applicable to all these indeterminate forms.