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

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

The Mean Value Theorem by Cauchy is the basis for proving a number of important results in Calculus. This theorem states that if f is a function defined on an interval [a, b], then there exists at least one point c in [a, b] such that f(c) equals the average of f(a) and f(b). Hence, we can use this theorem to find integrals and derivatives of functions defined on an interval.

Cauchy is a French mathematician living in the 19th century. That man made many important contributions to mathematics, including a theorem that states that if f and g are two functions continuous on the closed interval a,b and differentiable on the open interval a,b and if g'(x) > 0 for all x in the open interval (a,b), then this ratio: f(b) - f(a)/g(b) - g(a), you can think of the relationship between the derivatives f' and g' as f' - c being equal to f - c/g at some point c in the interval a,b.

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We'll see later on in Chapter Four that this is actually a special case of a Mean Value Theorem, so we'll make use of Cauchy's version to prove L'Hopital's Rule.

Let the ratio of f(x) over g(x) be considered and its limit sought as x tends to a. We will then consider two cases. In the first case, x is on the left of a; in the second case x is on the right of a.

Now by the Cauchy Mean Value Theorem, we saw earlier that this ratio is equal to this because f(a) and g(a) are equal to zero. Okay, these are just zero. Using Cauchy's Mean Value Theorem, we can write this ratio as f-dash c divided by g-dash c for certain number c between x and a.

Remember that x is on the left-hand side of a and c is somewhere between.

Now, we assumed in L'Hopital's Rule that the ratio f/g has a limit when c tends to a. In other words, both left-hand limit and right-hand limit exist and equal to the limit when c tends to a-minus and when c tends to a plus.

If x is on the left-hand side, let x tend to a. Since c is between x and a, c must also tend to a on the left-hand side same way.

If you consider x tends to a-minus of this ratio, then it is equal to this ratio when c tends to a-minus.

Let's suppose that x lies on the left side of a. In a similar way, you can consider the situation when x is on the right side of a. So if you equate this to this equality, you get x tends to a from the right side of this ratio equals the limit as c tends to a from the right side of the same ratio.

These two things are equal to each other and this thing.

Thus, the two numbers on the left-hand side of this equation must also be equal to each other and equal to that.

When two numbers are equal to the limit x tends to a of this ratio and the two numbers are equal to each other, then they must be equal to the limit x tends to a.