Growth rates of sequences
Use Theorem 10.6 to find the l...

Question

Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.

aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)

Accepted Answer

Welcome back, everyone. Let C and be equal to 5 power of N plus n the power of 200, all divided by 5 of N + 4 of N. Find limit as N approaches infinity of CN A 0 B 1, C2, and D infinity. For this problem, let's go ahead and write down the limit as N approaches infinity of 5 to the power of N plus N to the power of 200 divided by 5 of N plus 4 N. If we simply attempt and substitute an approaches infinity, we're going to get an infinite numerator and an infinite denominator, and infinity divided by infinity is an indeterminate form. So what we can do is simply divide all of our terms by 5 power of N. We can see that this is an exponential term that is repeated, and we're going to get 1 plus N to the power of 200 divided by 5 to the power of N, all divided by 1 plus. For it's the power of n divided by 5 to the power of n. And now using the properties of exponents, we can rewrite our last term as 4/5. Raise the power of N. And now let's analyze the behavior of those resultant fractional terms. N to the power of 200 divided by 5 to the power of n is going to approach 0 as n approaches infinity because our numerator contains a polynomial function and our denominator contains an exponential function. Let's recall that exponential functions grow faster than any polynomial functions as an approaches infinity. And now considering 4 divided by 5 raises the power of N, it is going to approach 0 as N approaches infinity because the absolute value of 4/5 is less than 1. So we are raising a fractional number that is less than 1 to the power of N wheren approaches infinity, right? So essentially it will approach 0. And we get our limit of one divided by one, which is one. Therefore, the correct answer for this problem corresponds to the answer choice B1. Thank you for watching.

Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.

aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)

Key Concepts

Limits of Sequences

Dominant Term and Growth Rates

Theorem 10.6 (Limit of Ratio of Sequences)

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