Hi folks. I'm stuck on evaluating a limit. Any help would be appreciated.
Since the limit produces the form , applying L'Hopital's rule gives the reciprocal function, which in turn is of the form . Applying L'Hopital's rule again gives the original function. Help!
You have the classic problem and the answer is...the question doesn't make sense.
Think about. This notion you have of the function is only defined in terms of real numbers. Is ? No, so it's not a debate of "is infinity so big it makes the zeroth power go away or is the zero so small it makes the infinity go away" the answer is you really can't ask the question without introducing some new number and doing this (at least in th obvious way) screws up the algebraic structure of ( and since this isn't true we can't additively cancel things). So, if you're asking what is where the answer is...it can be any real number you want.
Firstly, the difference in subtetly of the definitions is astounding. If you have any knowledge of complex analysis you should know this well considering for example the difference between and . Secondly, there is a fine difference. The point here (that I am concerned with) is an algebraic one. As it stands is a complete ordered field and in fact the only one up to isomorphism and so unless you want to completely redefine many aspects of our number system which we may take for granted you'll have to define in a way which makes it essentially the same as .
For example as I said you always here but then wouldn't even be a group.
Now, you can append the point " " for topological purposes (this is the Alexandroff compactification of locally compact Hausdorff spaces) but you'll get the unit circle
I think I'm rambling now...sorry it's late.