# Stuck with this limit of a sum: $\lim _{n \to \infty} \left(\frac{a^{n}-b^{n}}{a^{n}+b^{n}}\right)$.

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.

• Dec 29, 2016 at 11:02

If $b > a$, divide both the numerator and denominator by $b^n$ to get: $$\lim_{n \to \infty} \frac{\frac{a^n}{b^n}-1}{\frac{a^n}{b^n}+1}=\frac{-1}{1}=-1$$
• @Euler_Salter Why so? When $a < b$, we have that $\frac{a^n}{b^n} \to 0$ as $n \to \infty$ and thus the limit is just $\frac{0-1}{0+1}$. Dec 29, 2016 at 1:22
• oh thats right too .when u divide the second result i mentioned by $\frac{b^{n}}{a^{n}}$ u ll get the same equation u wrote .. damn maths are so fascinating and thanks btw Dec 29, 2016 at 1:25
Hint: In case of $b>a$ you divide by $b^n$ In case of $a=b$ it is simply zero.