Proof of limit with the nth root of x

Epsilon delta proof that the limit, as x goes to a, of the nth root of x, is a.

NOTE: There is an error at 7:42. Instead of a^(1/n), it should be a^((n-1)/n). Fortunately, the proof still works with a^((n-1)/n). Simply replace all instances of a^(1/n) with a^((n-1)/n), and you are good.

[SimchaWaldman]

Is this proof relying on the fact that the function f(x) = xⁿ is defined and continuous, for a^((n-1)/n)?
Or is it independent of that?

[egemertcomert]

how can we prove it when a<0 and n is a odd integer of course

[WillNewton]

at 7:42 why is it a^(1/n) ?
why not a^((n-1)/n)?