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.

{√(n² + 1) − n}

Accepted Answer

Welcome back every once another video. Evaluate the limit of the sequence Bn defined by BN equals square root of N2 + 4 N minus N as N approaches infinity or state if the sequence diverges. A 0 B 1, C2 and D diverges. So for this problem, we're going to write the limit as N approaches an infinity of BM. And this is going to be the limit as N approaches infinity of square root of N2 + 4 N. Minus M If we simply attempt and substitute and approaches infinity, we're going to get infinity minus infinity, which is an indeterminate form. So what we're going to do is simply multiply and divide by the conjugate of that term. So let's go ahead and rewrite it. We have square root of N2 + 4 N minus N, and we're going to multiply by the conjugates. So we're going to replace that negative sign with the positive sign. We get square root of N2 + 4 N + 4. And I'm sorry, plus M, and we have to divide by the same term to ensure that our expression is unchanged. So we're dividing by square root of N2 + 4 N + M. Now, let's apply the difference of squares factorization formula in reverse, right? Because we have a factored form. So multiplying, we get N2 + 4 N minus N2. And we're dividing by square root of N2 + 4 N + n. Let's notice that we can cancel out the N squared terms, and then once again, we're going to get an indeterminate form infinity divided by infinity if N approaches infinity. So, what we can do is simply divide both sides by N. Before we do that, we're going to write limit as and approaches infinity. We can factor out 4, and we have N in the numerator. Considering our denominator, we can basically take out N squared right from our square root. Notice that N is going to be greater than 0 in this case because this approach is positive infinity. So if we take out square root of N2, this is going to be N, right? In general, it's the absolute value of N, but in this case N is positive, so we're going to take out N, which leaves us with square root of N2 divided by N2 is 1, and 4 N divided by N2 is 4 divided by N. and we're adding N. And now let's go ahead and divide both sides by N. We're going to have 4 multiplied by a limit, as N approaches infinity of 1. Divided by square root of. 1 + 4 divided by n + 1. Now notice as n approaches infinity, our fractional term approaches zero. Because we're dividing 4 by an infinitely large number. And then we can calculate the limit, which is 4 multiplied by 1 divided by a square root of 1 + 1, and that's 4 divided by 2, which is 2. So the final answer for this problem corresponds to the answer to a C. Thank you for watching.

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

{√(n² + 1) − n}

Key Concepts

Limit of a Sequence

Algebraic Manipulation for Limits

Behavior of Functions as n Approaches Infinity

Watch next