TITLE: Stuck with this limit of a sum: $\lim _{n \to \infty} \left(\frac{a^{n}-b^{n}}{a^{n}+b^{n}}\right)$.
QUESTION [6 upvotes]: Here's the limit: $$\lim _{n \to \infty}  \left(\frac{a^{n}-b^{n}}{a^{n}+b^{n}}\right)$$
The conditions are $b>0$ and $a>0$.
I tried this with the case that $a>b$:
$$\lim _{n \to \infty}  \left(\frac{1-\frac{b^{n}}{a^{n}}}{1+\frac{b^{n}}{a^{n}}}\right)$$
It gives me the result $1$.
But, in the case of $b>a$, I don't find a solution. Thanks for your attention.

REPLY [7 votes]: If $b > a$, divide both the numerator and denominator by $b^n$ to get:
$$\lim_{n \to \infty} \frac{\frac{a^n}{b^n}-1}{\frac{a^n}{b^n}+1}=\frac{-1}{1}=-1$$

REPLY [4 votes]: Hint: In case of $b>a$ you divide by $b^n$
In case of $a=b$ it is simply zero.