Why does l'hopital's rule work?

[Terrell]

What is the intuition behind it? when i watch videos of people using l'hopital's rule. i can only deduce that they're only taking derivatives over and over again until a number comes out and that becomes the limit. how can a tangent slope be a value for a limit? Please give me an intuitive explanation. I'm a novice with abstract explanations. thank you!

 

[Ssnow]

If you look at the Cauchy theorem

https://en.wikipedia.org/wiki/Mean_value_theorem#Cauchy.27s_mean_value_theorem

this can give you some idea how derivatives and the increments of two functions are related .. just for the intuition there is point in which the ratio of the derivatives is the ratio of the two increments of ##f## and ##g## ... I don't know if this can help you ...

 

[jedishrfu]

Here's a discussion of it at Math Stack Exchange:

http://math.stackexchange.com/questions/98082/why-does-lhospitals-rule-work

One poster uses a simple example of evaluating the limit a fraction and how the rule helps. (see the 2nd block of posts)

 

[PeroK]

If ##f(a) = g(a) = 0## then it's as simple as:

##lim_{x \rightarrow a} \frac{f(x)}{g(x)} = lim_{x \rightarrow a} \frac{(f(x) - f(a)/(x-a)}{(g(x) - g(a))/(x-a)} = \frac{f'(a)}{g'(a)}##

That's the essence of it, anyway.

 

[Terrell]

If ##f(a) = g(a) = 0## then it's as simple as:

##lim_{x \rightarrow a} \frac{f(x)}{g(x)} = lim_{x \rightarrow a} \frac{(f(x) - f(a)/(x-a)}{(g(x) - g(a))/(x-a)} = \frac{f'(a)}{g'(a)}##

That's the essence of it, anyway.

wow! fascinating! lol...

 

[BreCheese]

If ##f(a) = g(a) = 0## then it's as simple as:

##lim_{x \rightarrow a} \frac{f(x)}{g(x)} = lim_{x \rightarrow a} \frac{(f(x) - f(a)/(x-a)}{(g(x) - g(a))/(x-a)} = \frac{f'(a)}{g'(a)}##

That's the essence of it, anyway.

This formula is intuitive and really drives home what a derivative actually is. Thanks for sharing it.
f(x) - f(a) is really a fancy way of saying change of y, denoted as Δy, or in other words, y2 - y1. Similarly, x - a is a fancy way of writing change in x, denoted Δx, or in other words x2 - x1.
As "x" approaches "a" the change in x gets smaller, and so does the change in y. When we had to find slopes in algebra we had to use a point-slope form, which was (y2-y1) / (x2-x1). We can apply this same concept to non-linear functions, and when we make "x" really close to "a" then we get a more accurate approximation of the slope of the curve. Furthermore, a first derivative is the SLOPE of the line tangent to a function, and where this tangent line touches the function the two functions are said to have the same slope. In other words, when a=x we then have an instantaneous rate of change; a derivative. In short, I love all the implications from this formula.

I am not sure why L'Hopital's rule works, but I do know that you must first observe a limit to yield an indeterminate form (0/0, ∞/∞, etc) before you can apply L'Hopital's rule. The formula PeroK provided is interesting because it makes a ratio of the instantaneous rate of change of function f to the instantaneous rate of change of function g at the same point "a". Maybe because f(x) divided by g(x) is a ratio, looking at the slopes of each function (and the ratio of the slopes at a particular "a" value) can provide logical insights. When I think of L'Hopital's Rule, I can't help but think of Taylor series (which is a summation of derivatives (of derivatives) and corresponding polynomials, which approximate a given function at a value x near the center of the function). Although a strange thought, can anyone relate Taylor series to L'Hopital's Rule?