L Hospital Rule Explained: Origins & Meaning

[muzz]

can somebody explain what is the L hospital rule.. where does it come from and what's it mean?

 

[J77]

Sometimes it's easier to look at the limit of the derivatives of the functions, this is basically what you use it for: http://mathworld.wolfram.com/LHospitalsRule.html

 

[nicktacik]

1)Express $$f(x)$$ as $$\frac{g(x)}{h(x)}}$$

2)Is $$\lim_{x\rightarrow x_0}\frac{g(x)}{h(x)}=\frac{0}{0}$$ or $$\frac{\infty}{\infty}$$ ?

3) If so, $$\lim_{x\rightarrow x_0}\frac{g(x)}{h(x)} = \lim_{x\rightarrow x_0}\frac{g'(x)}{h'(x)}$$

 

[mathwonk]

think of the case where g,h are given by power series centered at x0. then its obvious.

 

[GoldPheonix]

can somebody explain what is the L hospital rule.. where does it come from and what's it mean?

L'Hopital's (Low-pi-tal; tal like tally without the y sound) Rule* is a way to resolve limits of a function when by simple direct substitution fails. [IE: given the limit "lim(x->n) of f(x)", direct substitution is when you plug in n for x and solve it down. If this yields 0/0 or infinity over infinity, you need L'Hopital's Rule to resolve the conflict].

What it does instead of directly inputting values is look at what "affect" the values have on the denominator and numerator in terms of which one is growing faster as the values change (mostly, as they ascend to infinity or descend to zero). Look at the following function:

$$f(x) = \frac{5^x}{x^2}$$

If (if is important here) as 5^x approaches infinity, it is way more than x^2, then the whole function will get bigger, right? I mean, look at a function's denominator and numerator not in turns of shape, but in terms of quantity; how much "bigger" or "smaller" the other one is as x changes. If something is getting bigger on the bottom than it is on the top, it'll get smaller. If it's visa versa, it'll get bigger. That's all L'Hopital's Rule is stating in mathematical form; it is meant to help with the problems that arise as we have to deal with infinity.

How do we figure which one is increasing faster than the other at an instantaneous level? With the derivative slope, of course. If I'm growing at a faster rate than the other guy, at some point I will surpass that guy and then just keep on getting bigger until I "swallow" him up so to speak.

So, all L'Hopital's Rule means is this:

$$lim(x->h) f(x) = \frac{N(x)}{D(x)} = \frac{N'(x)}{D'(x)} = \frac{N''(x)}{D''(x)}...$$

(Where N(x) is the numerator, and D(x) is the denominator)

NOTE: They only work until you can plug in a singular value (or right when you get to none), otherwise you'll get incorrect values if you differentiate after that. Good rule of thumb is only to differentiate as needed, then use that value.

So, the uses of L'Hopital's Rule are to find limits when you plug in infinity, to find limits that naturally involve infinity (lim[x^2/(x-1)] at "x=1"), and, I can't figure out why yet, just limits in general. It works, so I won't complain. =P