L'Hopital's rule for increasing functions

[fakecop]

L'Hopital's rule greatly simplifies the evaluation of limits of indeterminate forms, especially those with polynomial terms. This is because every time you take the derivative of a polynomial, the exponent decreases by 1, until it becomes a constant function, at which point the limit can be evaluated.

So the question is, how can you apply L'Hopital's rule to an indeterminate form such as e^x/x^x? or x^x/x!? for some functions, such as the double exponential, factorial, and hyperoperations (tetration, pentation...) each successive ordered derivative increases everytime you differentiate.

So is there a way to resolve this problem using the tools of elementary calculus? I know that for series there are tests to test convergence, but is there a similar procedure for continuous functions?

 

[lurflurf]

What we are trying to do with L'Hopital's rule is not reduce the degree of the polynomial though that happens sometimes, we are trying to reduce the limit so that it is not an indeterminate form.

Here is a humorous example
$$\lim_{x \rightarrow 0} \frac{\sinh \, \tanh \, x-\tanh \, \sinh \, x}{\tan \, \sin \, x-\sin \, \tan \, x}=1$$

a more manageable example is
$$\lim_{x \rightarrow 0} \frac{\tan \, x}{\sin \, x}=1$$

 

[chiro]

Hey fakecop.

There are different ways to look at problems like this.

One way that pops up (for your problem) is to do a taylor series expansion of e^x and then divide each term by x^x.

This means that if you have a term a_n*x^n / x^x, then you will have a_n * x^(n-x).

If you then take the limit of each term (since x > n as x -> infinity) then you can show that each term goes to 0 and thus the limit goes to 0 as well.

Other techniques include using norms and bounds to show that a limit is bounded by some expression (possibly with an error term) attached. This idea is used in things like Euler-Mclaurin formula where you want to get a relationship between sums and integrals and is used when integrating the integral is easy, but the sum is not (in terms of an analytic evaluation).

In short, you basically think of ways to take the expression and transform it into something else that is exact or a good enough approximation, and use these facts to see if you can use that expression or transform it into something else that is manageable (and so on) by going from one decomposition to another.