Question

I need to prove the limit as x approaches 1 of x^(1/2)=1. In order to i have have to figure out how to factor x^(1/2)-1. I am stuck at factoring it.

Answer 1

why?

Answer 2

i mean why do you have to factor?

Answer 3

to prove that lx^(1/2)-1l< eplsilon when delta>lx-1l.
\[\to prove that \left| \sqrt{x}-1 \right|<\epsilon when \left| x-1 \right|< \]

Answer 4

you could try saying, for example, that since x is approaching 1, it is certainly less than 4 for \(\sqrt{x}+1<3\) giving you
\[|\sqrt{x}-1|<\epsilon \]
\[|\sqrt{x}-1||\sqrt{x}+1|<3\epsilon\]
\[|x-1|<3\epsilon\] so you can pick a \(\delta=\frac{\epsilon}{3}\) and run the argument backwards as usual

Answer 5

I will try that thank you