<s>[INST] Help thorougly proving a limit: $a>b>0; (a^n-b^n)/(a^n+b^n)$ as $n\to \infty$ Clearly, using basic calculus, I can find that the limit is 1. This is what I had so far:
$(1)$ $\frac {a^n-b^n}{a^n+b^n} = (2)$ $\frac {a^n}{a^n+b^n} - \frac {b^n}{a^n+b^n}
= (3)$ $\frac {1}{1+ \frac{b^n}{a^n}}-\frac {1}{1+ \frac{a^n}{b^n}}$
$(4)$ $a>b \implies (5)$ $\frac{b^n}{a^n}=0$ and $(6)$ $\frac{a^n}{b^n}=\infty$ as $n \to \infty$.
$(7)$ $\frac {1}{1+ \frac{b^n}{a^n}}-\frac {1}{1+ \frac{a^n}{b^n}} = (8)$ $\frac {1}{1+0} - \frac {1}{1+\infty} = (9)$ $1-\frac{1}{\infty}=(10)$ $1-0=1$
I asked another student whether what I had was correct; he said it was correct, however, I should put an indication of a theorem or definition at each step that I make.
So far I have a theorem that says I can use the step that the sum of limits is equal to the limit of sums. Some steps are trivial, but I'm not sure which ones are. Are there any more theorems or definitions I can use to identify what steps I'm taking?
 [/INST] If you mean that $n\rightarrow+\infty$ then
$$\frac{a^n-b^n}{a^n+b^n}=\frac{1-\left(\frac{b}{a}\right)^n}{1+\left(\frac{b}{a}\right)^n}\rightarrow\frac{1-0}{1+0}=1.$$
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