Expert: Paul Klarreich - 11/6/2006

Question
I want the proof of Power rule in limit laws which states that the limit of the rational power of a function is the rational power of the limit of the function.

Answer
Questioner:  Minirani
Category:  Calculus
 
Question:  I want the proof of Power rule in limit laws which states that the limit of the rational power of a function is the rational power of the limit of the function.
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Hi, Minirani,

I'm not sure of the context of the question (is this elementary calculus, advanced analysis, measure theory???) so I'll try this approach and see if it fits:

You can use these known limit properties for functions: (to save typing, I leave out the phrases "of a function" or "of two functions", etc.)

A. The limit of a sum (or difference) is the sum (or difference) of the limits.
B. The limit of a product is the product of the limits.
C. The limit of a quotient is the quotient of the limits, if it exists.
D. The limit of a constant is the constant.

Formally, those (and the ones that are coming up) are written like this: (I will also leave out  'x --> a' to reduce the typing.)

A. lim[f(x) + g(x)] = lim f(x) + lim g(x)

C. lim [f(x)/g(x)] = lim f(x)/lim g(x), provided  lim g(x) /= 0   << not equal to

Then we can use B, repeatedly to prove that:

E. For any positive integer n,

lim [f(x)^n] = [lim f(x)]^n

E. The limit of a positive integral power is the power of the limit.

Now it is also possible to prove, for any positive integer n, that:

F. The limit of the n-th root of a function is the n-th root of the limit.

[It's hard to write roots on the computer, so I will use the 1/n-th power instead.]

F. lim [f(x)^(1/n)] = [lim f(x)]^(1/n), PROVIDED lim f(x) > 0

Now you have all you need for

G. The limit of a rational power is the rational power of the limit.

G. lim [f(x)^(m/n)] = [lim f(x)]^(m/n), provided  lim f(x) > 0

To prove G, you write:

lim [f(x)^(m/n)]
= lim [ [f(x)^(1/n)]^m] ] by exponent rules
= [lim [f(x)^(1/n)]^m,  by rule E
= [[lim f(x)]^(1/n)]^m, provided lim f(x) > 0, by rule F
= [lim f(x)]^(m/n)  by exponent rules

Now you might need proofs of some of these rules; if that's the case, let me know.