13–52. Limits of sequences
Find the limit of the followi...

Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(√(4n⁴ + 3n))⁄(8n² + 1)}

Accepted Answer

Welcome back everyone. Determine the limit as n approaches infinity of the sequence DN equals square root of 16 to the power of 4 minus 8 N divided by 20 and 2 + 4 or state if it diverges. A 0 B 1/5 C 2/5, and D diverges. For this problem, let's go ahead and write down the limit. Limit as N approaches infinity of the subscript N. So that's going to be square root of 16 and raise the power of 4 minus 8N. Divided by 20 and 2 + 4. If we simply attempt and substitute infinity for and we're going to get an indeterminate form, that's going to be square root of infinity minus infinity divided by infinity. So what we can do is simply perform factorization. Starting with the numerator, right? And what we're going to do is simply factor out and raise the power of 4. We're choosing the highest power of M. So we're going to get limit as N approaches infinity of square root of and raise the power of 4 in parent we get 16 and I'm sorry, we get 16, right, because we're factoring out and the power of 4 minus. 8 Divided by n cubed. And now our denominator contains 20 and 2 + 4. Remember that N is greater than or equal to 1 because it is a sequence where N begins with 1, right? So we're considering positive values of N, and what we're going to do is simply take square root of N erase the power of 4, which is going to be positive and quad. So we get a limit as N approaches infinity of N2d multiplied by square root. Of 16 minus 8 divided by n cubed all divided by 20 and 2 + 4. And then we can divide both sides by N2d, right? So we're going to cancel out N2d in the numerator, and for the denominator, we can just add. Divided by N2. So 20 and 2 is going to be divided by N2, and 4 is going to be divided by N2. Now, once again, 20 and 2 divided by N2 is going to be 20, and now we can evaluate the limit. All of our fractional terms, specifically in -8 divided by n cubed and 4 divided by N2d, are going to approach 0. Because those are constants divided by an infinitely large number, right? And now we can use the remaining terms. We have square root of 16. Divided by 20 + 0, or basically 20. So we get 4 divided by 20, which is equal to 1/5. Therefore, the correct answer for this problem is option B 1/5. Thank you for watching.

13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.

{(√(4n⁴ + 3n))⁄(8n² + 1)}

Key Concepts

Limits of Sequences

Asymptotic Behavior and Dominant Terms

Simplification Using Algebraic Manipulation

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