Limit of a Sequence (Updated with progress)

In summary, the conversation is about finding the limit of a sequence using the definition of the limit. The participant has figured out that the limit should be 1, but is not sure how to prove it. The expert suggests dividing the top and bottom of the fraction and using a theorem to show that the limit is indeed 1.
  • #1
slwarrior64
22
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Hello, I know I posted this question recently but I wanted to update with my progress. I have figured out what the limit should be but I would really appreciate help with how to use the definition of the limit of a sequence to prove it! What I have is:Suppose n is extremely large, then both n^2+1 and n^2 have almost the same size. Similarly both n and n+1 have almost the same size. Therefore, as n grows larger and larger, we can replace the numerator by sqrt (n^2)=n and the denominator by n. Therefore the guess value for the limit should be 1 since the numerator and denominator should cancel.

I am pretty sure this is right, but I am not sure how to answer it in the way this question is asking.
 
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  • #2
slwarrior64 said:
View attachment 11131
Hello, I know I posted this question recently but I wanted to update with my progress. I have figured out what the limit should be but I would really appreciate help with how to use the definition of the limit of a sequence to prove it! What I have is:Suppose n is extremely large, then both n^2+1 and n^2 have almost the same size. Similarly both n and n+1 have almost the same size. Therefore, as n grows larger and larger, we can replace the numerator by sqrt (n^2)=n and the denominator by n. Therefore the guess value for the limit should be 1 since the numerator and denominator should cancel.

I am pretty sure this is right, but I am not sure how to answer it in the way this question is asking.
That is definitely the right idea. The way to express it in more formal mathematical language is to divide top and bottom of the fraction by $n$ so as to get $$a_n = \frac{\sqrt{1 + \frac1{n^2}}}{1 + \frac1n}.$$ The numerator and denominator of that fraction both go to $1$ as $n\to\infty$, and you can use the theorem that the limit of a quotient is the quotient of the limits: $$\lim_{n\to\infty}\frac{\sqrt{1 + \frac1{n^2}}}{1 + \frac1n} = \frac{\lim_{n\to\infty}\sqrt{1 + \frac1{n^2}}}{\lim_{n\to\infty}\left(1 + \frac1n\right)} = \frac11 = 1.$$
 
  • #3
Opalg said:
That is definitely the right idea. The way to express it in more formal mathematical language is to divide top and bottom of the fraction by $n$ so as to get $$a_n = \frac{\sqrt{1 + \frac1{n^2}}}{1 + \frac1n}.$$ The numerator and denominator of that fraction both go to $1$ as $n\to\infty$, and you can use the theorem that the limit of a quotient is the quotient of the limits: $$\lim_{n\to\infty}\frac{\sqrt{1 + \frac1{n^2}}}{1 + \frac1n} = \frac{\lim_{n\to\infty}\sqrt{1 + \frac1{n^2}}}{\lim_{n\to\infty}\left(1 + \frac1n\right)} = \frac11 = 1.$$


That makes sense, Thanks!
 

Related to Limit of a Sequence (Updated with progress)

1. What is a limit of a sequence?

A limit of a sequence refers to the value that a sequence approaches as the number of terms in the sequence increases. It is the value that the terms in the sequence get closer and closer to, but may never actually reach.

2. How is the limit of a sequence calculated?

The limit of a sequence is typically calculated by observing the pattern of the sequence and determining the value that the terms seem to be approaching. This can also be done using mathematical techniques such as the squeeze theorem or the ratio test.

3. What is the importance of the limit of a sequence?

The limit of a sequence is important because it helps us understand the behavior of a sequence as the number of terms increases. It also allows us to make predictions about the behavior of the sequence in the long run.

4. Can a sequence have multiple limits?

No, a sequence can only have one limit. However, a sequence can have a limit that does not exist, which means that the terms in the sequence do not approach a specific value as the number of terms increases.

5. How is the concept of limit of a sequence related to calculus?

The concept of limit of a sequence is closely related to calculus as it is a fundamental concept in the study of limits and continuity. It is also used in the definition of derivatives and integrals, which are important concepts in calculus.

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