# Does $\lim_{n\to\infty}\frac{3^n+5^n}{(-2)^n+7^n}$ exist?

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?

• The limit is indeed zero, so it's likely a typo. – user296602 Dec 3 '18 at 16:38
• You’re correct. – KM101 Dec 3 '18 at 16:39

$$\frac{3^n+5^n}{2^n+7^n}\le \frac{3^n+5^n}{(-2)^n+7^n}\le \frac{3^n+5^n}{-(2^n)+7^n}$$
and refer to squeeze theorem or more simply dividing by the by leading term $$7^n$$
$$\frac{3^n+5^n}{(-2)^n+7^n}=\frac{(3/7)^n+(5/7)^n}{(-2/7)^n+1}$$