Properties of Limits:Lim 2^(1/n) = 2^0 = 1

[tarheelborn]

Homework Statement

What property of limits says that lim 2^(1/n) = 2^lim (1/n) = 2^0 = 1? Thanks.

Homework Equations

The Attempt at a Solution

 

[Dick]

It's not a property of limits. It's a property of the function f(x)=2^x. f(x) is continuous at x=0.

 

[tarheelborn]

I don't follow that. Because 2^x is continuous at x = 0, this means that lim 2^x = 2 ^ lim x?

 

[Dick]

f(x) is continuous at x=a means lim x->a f(x)=f(a). That's the definition of continuity. Apply it to f(x)=2^x and a=0.

 

[tarheelborn]

Oh, duh... Thank you so much.

 

[HallsofIvy]

But there is a "law of limits" involved:

If $\lim_{x\to a} f(x)= L$ and $\lim_{n\to\infty} x_n= a$ then $\lim_{n\to\infty} f(x_n)= L$.