Limit of a sequence. Hello friends, today I will talk about the limit of a sequence. Have a look!!

**Limit of a sequence – some rules**

In my earlier posts, I have talked about different kind of sequences like arithmetic sequences, geometric sequences and so on. Now I’ll show how to determine the limit of any sequence.

Ok, the limit of a sequence follows some rules like anything else.

Suppose, and are two sequences.

So the **rules** are as follows:

- for , the limit of the sequence will be
- if any sequence looks like , the limit of the sequence will be
- for a product of two sequences and , the limit will be
- if any sequence looks like , the limit of the sequence will be
- for a sequence , the limit can be different depending on the value of as

Now I’ll show some examples where I have to use these rules.

**Limit of a sequence – relevant examples**

*Disclaimer: None of these examples is mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.*

So here are the examples.

**Example 1**

According to Stroud and Booth (2013)* “Evaluate the following limit ”

**Solution**

Here I have to find out the limit of the sequence when tends to be infinity.

Since the sequence is a sum of two sequences and respectively, I’ll use the rule no. 1 (from above) to find out the limit.

(1)

First of all, I’ll find out the limit of the sequence .

(2)

Next, I’ll find out the limit of the sequence .

Thus I can write as

So the limit of the sequence is

Hence the limit of the sequence will be

(3)

Now this happens because any number multiplied, divided, added or subtracted from infinity becomes infinity.

Therefore I can substitute the values from equations (2) and (3) to the equation (1) to get the limit as

Hence the limiting value of the sequence is . That is the answer to this example.

Here comes my next example.

**Example 2**

According to Stroud and Booth (2013)* “Evaluate the following limit ”

**Solution**

Here I have to find out the limit of the sequence when tends to be infinity.

First of all, I’ll simplify the sequence .

Thus it will be

So I can say

Now I’ll use rule no. 5 (from above) to find out the limit. According to this rule, for any value of , the limit is infinity.

Thus, in this case, it will be

Hence my conclusion is is the solution to this example.

Now I’ll give another example.

**Example 3**

According to Stroud and Booth (2013)* “Evaluate the following limit ”

**Solution**

Here I have to find out the limit of the sequence when tends to be infinity.

First of all, I’ll use rule no. 2 (from above) to find the limit.

So it will be

Now, just like the first example, will be

Next, for the second part, that is, I’ll use rule no. 5 to find the limit.

According to this rule, for , the limit is undefined.

Thus will also be

Therefore I can say

Hence my conclusion is is the solution to this example.

Now I’ll give another example.

**Example 4**

According to Stroud and Booth (2013)* “Evaluate the following limit ”

**Solution**

Here I have to find out the limit of the sequence when tends to be infinity.

First of all, I’ll simplify the sequence .

Thus it will be

So I can say

Next, I’ll use rule no. 5 (from above) to find the limit.

According to this rule, for , the limit is .

Now in this case .

Thus will be

Hence I can say that this is the solution to this example.

Now I’ll give another example.

**Example 5**

According to Stroud and Booth (2013)* “Evaluate the following limit ”

**Solution**

Here I have to find out the limit of the sequence when tends to be infinity.

First of all, I’ll use rule no 4 (from above) to find out the limit.

Thus it will be

When tends to be infinity, tends to be zero. So I’ll bring in both at the numerator and denominator.

Now I can see the highest power of in this fraction is 2.

So I’ll divide both the top (numerator) and the bottom (denominator) of this fraction with to get

Therefore the limit will be

Hence my conclusion is is the solution to this example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

Top Maritime universities says

Great article about RULES on the LIMIT OF A SEQUENCE…