Solve Limits of Sequence: Detailed Instructions

In summary: You can write the code directly in the forum by using the MATH button on the toolbar. It will generate the [MATH] tags for you. As for the third problem, try factoring first and then rationalizing the numerator, as follows:n\sqrt{n^2+4}-n^2=n\left(\sqrt{n^2+4}-n \right)\cdot\frac{\sqrt{n^2+4}+n}{\sqrt{n^2+4}+n}=?Once you do this then divide each term by...?I hope the forum format won't get on your nerves...its just that i am not very good with latex,so i needed to write in forum in such a way
  • #36
(Ok, here is my last post on this for now. I think you need to see the next few steps written out so here they are plus some advice)

I think you might want to practice some easier problems first for using these methods. This one has a lot of parts to it if you're seeing it for the first time. Usually limits are given where you have rationalize a fraction somehow and then everything will cancel nicely and easily. After that the cancellation might be trickier. Finally limits like #3 are given and it can be messy.

Being able to quickly choose the proper fraction to multiply with the original limit is key. It is usually going to be the conjugate of the numerator or denominator. Here are some examples.

This is where we are at for #3 so far:

\(\displaystyle \frac{n\left(\sqrt{n^2+4}-n \right) \left(\sqrt{n^2+4}+n\right)}{\sqrt{n^2+4}+n}=\frac{4n}{\sqrt{n^2+4}+n}\)

From here you need to divide all terms by $n$. That means evaluate
\(\displaystyle \frac{4n}{n}, \frac{\sqrt{n^2+4}}{n},\frac{n}{n}\)
For
\(\displaystyle \frac{\sqrt{n^2+4}}{n}\) you can use this property: \(\displaystyle \frac{\sqrt{n^2+4}}{n}=\sqrt{\frac{n^2+4}{n^2}} =\sqrt{ \frac{n^2}{n^2}+\frac{4}{n^2}}\)

You hopefully see now why algebra is so important here. Review these concepts some and practice a few easier problems to get a feel for the process. Once you have that you can do this problem in 1-2 minutes probably by knowing the methods you'll probably use.
 
<h2>1. What is a limit of sequence?</h2><p>A limit of sequence is a mathematical concept that describes the behavior of a sequence as its terms approach a certain value. It is the value that the terms of the sequence get closer and closer to, but may not necessarily reach.</p><h2>2. How do you solve limits of sequence?</h2><p>To solve limits of sequence, you need to understand the behavior of the sequence and use mathematical techniques such as the squeeze theorem, ratio test, or comparison test. These techniques can help determine the limit of a sequence by analyzing its terms and their relationships.</p><h2>3. What is the importance of solving limits of sequence?</h2><p>Solving limits of sequence is important in many areas of mathematics, such as calculus, analysis, and number theory. It allows us to understand the behavior of sequences and make predictions about their values, which can be applied to real-world problems and mathematical proofs.</p><h2>4. What are some common mistakes when solving limits of sequence?</h2><p>Some common mistakes when solving limits of sequence include not considering the behavior of the sequence as a whole, using incorrect mathematical techniques, or not checking for conditions that may affect the value of the limit, such as oscillation or divergence.</p><h2>5. Can limits of sequence have multiple solutions?</h2><p>Yes, limits of sequence can have multiple solutions, depending on the behavior of the sequence. Some sequences may have a finite limit, while others may have an infinite limit or no limit at all. It is important to carefully analyze the sequence and use the appropriate techniques to determine the correct solution.</p>

FAQ: Solve Limits of Sequence: Detailed Instructions

1. What is a limit of sequence?

A limit of sequence is a mathematical concept that describes the behavior of a sequence as its terms approach a certain value. It is the value that the terms of the sequence get closer and closer to, but may not necessarily reach.

2. How do you solve limits of sequence?

To solve limits of sequence, you need to understand the behavior of the sequence and use mathematical techniques such as the squeeze theorem, ratio test, or comparison test. These techniques can help determine the limit of a sequence by analyzing its terms and their relationships.

3. What is the importance of solving limits of sequence?

Solving limits of sequence is important in many areas of mathematics, such as calculus, analysis, and number theory. It allows us to understand the behavior of sequences and make predictions about their values, which can be applied to real-world problems and mathematical proofs.

4. What are some common mistakes when solving limits of sequence?

Some common mistakes when solving limits of sequence include not considering the behavior of the sequence as a whole, using incorrect mathematical techniques, or not checking for conditions that may affect the value of the limit, such as oscillation or divergence.

5. Can limits of sequence have multiple solutions?

Yes, limits of sequence can have multiple solutions, depending on the behavior of the sequence. Some sequences may have a finite limit, while others may have an infinite limit or no limit at all. It is important to carefully analyze the sequence and use the appropriate techniques to determine the correct solution.

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