$\lim_{n\to \infty} {a^n-b^n\over a^n+b^n}.$

Find the limit $$\lim_{n\to \infty} {a^n-b^n\over a^n+b^n}.$$ How do I find this limit? Thank you.

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Look at the possible cases $|a|=|b|$; $|a|<|b|$ or $|a|>|b|$.

The intuition here is that as $n$ grows, the larger of $|a|$ or $|b|$ will grow much faster than the smaller. It will come to dominate both the numerator and denominator, so you should be able to guess the limit by ignoring the smaller one. Once see the intuition and have a statement you want to prove, you should be able to find the limit using techniques you’ve seen before.