Let me state how I define exponentials.
Define
It is not hard to show that,
1) is an increasing function.
2) is differenciable and therefore continous for .
3) .
4)There exists a number, called , so that .
We see that is a one-to-one function. Define to be its inverse function on its range.
It is not hard to show that is increasing, differenciable and so continous, and furthermore, .
We have the following supprising property that if is a rational number then . So we define , to be . With that we generalize exponents as follows .
The difficutly besides for using the definition is to have a formal definition of what an exponent means.
Okay,
1)We define as the integral above.
2)Using FTC and other things we develop important properties for such as: .
3)I said that is continous and increasing as one of its properties so it has an inverse function which we define as .
Note, #3 does not have anything to do with from the way it is defined above.
4)It can be shown that if is a rational number then i.e. if then .
5)So we see that the function evaluated at is the same thing as raising the number (which is define to be so that ) to that power.
6)Now is seems "natural" to write rather as because for rational powers it makes sense, i.e. same as raising exponents. But for irrational powers (though they were still not define) this function still makes sense! Because is defined for all real .
I think this seems strange to you is that you never learned about what an "exponent" means. Yes -times. That is how we define them for positive integers. [math[a^0=1[/tex] that is a mere definition. But what about where is rational? You defined it as finding deminator roots. But how do you define where is any arbitrary number? That takes works. The approach above is the standard and simplest and nicest approach to this problem. Meaning, we define the exponent of to be provided that .
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My point is that the limit is hard to find because we never appropriately defined what means for .