- #1
Icebreaker
Prove using the def of lim that lim(x->a) x^n = a^n for all positive integers n.
The limit is obviously a^n. I've factored x^n - a^n and got
[tex] (x^{n/2}+a^{n/2})(x^{n/4}+a^{n/4})(x^{n/8}+a^{n/8})...(x+a)(x-a)[/tex]
Now, I have my x-a, which is needed for determining delta, i.e., |x-a|<d. However, that long chain of factors before it I have no idea how to get rid of. I could look for an upper bound, but it's unbounded. Can anyone help?
The limit is obviously a^n. I've factored x^n - a^n and got
[tex] (x^{n/2}+a^{n/2})(x^{n/4}+a^{n/4})(x^{n/8}+a^{n/8})...(x+a)(x-a)[/tex]
Now, I have my x-a, which is needed for determining delta, i.e., |x-a|<d. However, that long chain of factors before it I have no idea how to get rid of. I could look for an upper bound, but it's unbounded. Can anyone help?