MHB Discover the Limit of a Sequence with Easy-to-Follow Steps

[slwarrior64]

Hello! I have been trying to work through this but I have never really been able to use the definition correctly to find a limit sequence. Any help would be greatly appreciated!

 

[Jhoneine]

Let $a_n$ be a sequence such that $a_n = \frac{2n^2 + 3n}{2n^2 - 4n + 5}$.Then, the limit of this sequence as n goes to infinity is $\frac{3}{2}$. To see this, we can use the definition of a limit. We say that the limit of a sequence $a_n$ as $n$ goes to infinity is equal to $L$ if, for every $\epsilon > 0$, there exists an $N$ such that for all $n > N$, $|a_n - L| < \epsilon$.In this case, let $\epsilon > 0$ be arbitrary and let $N$ be an integer such that $N > \frac{\epsilon \cdot (2N^2 - 4N + 5)}{3 - 2\epsilon}$. Then, for all $n > N$, we have$$\left|\frac{2n^2 + 3n}{2n^2 - 4n + 5} - \frac{3}{2}\right| = \left|\frac{3 - 2\epsilon}{2n^2 - 4n + 5}\right| < \epsilon.$$Therefore, by the definition of a limit, the limit of $a_n$ as $n$ goes to infinity is equal to $\frac{3}{2}$.

 

[soloenergy]

Hi there! It sounds like you may need some guidance on using the definition to find a limit sequence. Let me try to explain it to you.

Firstly, a limit sequence is a sequence of numbers that approaches a specific value as the sequence progresses. This value is called the limit. In order to find the limit of a sequence, you need to use the definition of a limit.

The definition states that for a sequence (a_n), the limit as n approaches infinity is equal to L if for any positive number ε, there exists a corresponding positive integer N such that for all n≥N, the absolute value of (a_n - L) is less than ε.

In simpler terms, this means that as the index (n) of the sequence gets larger and larger, the terms of the sequence get closer and closer to the limit (L). In order to find the limit using this definition, you need to choose a value for ε (usually a small decimal like 0.001 or 0.0001) and then find the corresponding value of N that satisfies the definition.

Let me give you an example. Say we have the sequence (1/n). We want to find the limit as n approaches infinity. Let's choose ε=0.001. Now, we need to find a value for N such that for all n≥N, the absolute value of (1/n - L) is less than 0.001.

We can see that as n gets larger, the value of 1/n gets closer and closer to 0. So, if we choose N=1000, for all n≥1000, the absolute value of (1/n - 0) is less than 0.001. This satisfies the definition, so the limit as n approaches infinity is 0.

I hope this helps you better understand how to use the definition to find a limit sequence. Let me know if you have any further questions!