Basic real analysis - Root test (question on example from book)

Question

Just a quick question. I can't seem to see through the math or follow through with this example from the textbook. I don't see the connection or the jump where it says it concludes the limit is 1/2. It is the underlined part in the image.

Example from book, "Elementary Analysis, The Theory of Calculus

Answer 1

You have $2^{-1/n - 1} = {1\over2} 2^{-1/n}$ and $2^{1/n - 1} = {1\over2} 2^{1/n}$, giving you this limit.

Answer 2

Note

$$ (a_n)^{1/n} = 2^{ \frac{1}{n} - 1} = \frac{ 2^{\frac{1}{n}} }{2} $$

and

$$ 2^{ -\frac{1}{n} - 1} = \frac{ 2^{-\frac{1}{n}} }{2} $$