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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?

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    $\begingroup$ The limit is indeed zero, so it's likely a typo. $\endgroup$
    – user296602
    Dec 3, 2018 at 16:38
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    $\begingroup$ You’re correct. $\endgroup$
    – KM101
    Dec 3, 2018 at 16:39

1 Answer 1

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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 $7^n$

$$\frac{3^n+5^n}{(-2)^n+7^n}=\frac{(3/7)^n+(5/7)^n}{(-2/7)^n+1}$$

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