My textbook says that this limit doesn't exist, but I don't understand - why? I tried calculating it by taking from both numerator and denominator factors that diverge to $\infty$ the fastest:
$$\lim_{n\to\infty}\frac{3^n+5^n}{(-2)^n+7^n}= \lim_{n\to\infty}\frac{5^n}{7^n}=0$$
Did I do something wrong here?
Your result is correct but the way is not so much clear.
To solve properly we can observe that
$$\frac{3^n+5^n}{2^n+7^n}\le \frac{3^n+5^n}{(-2)^n+7^n}\le \frac{3^n+5^n}{-(2^n)+7^n}$$
and refer to squeeze theorem or more simply dividing by the by leading term $7^n$
$$\frac{3^n+5^n}{(-2)^n+7^n}=\frac{(3/7)^n+(5/7)^n}{(-2/7)^n+1}$$
