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How to find the limit sequence?

Aniket Singh , 5 Months ago
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Askiitians Tutor Team

Last Activity: 5 Months ago

To find the limit of a sequence, follow these steps:

1. **Understand the sequence**:
A sequence is a list of numbers arranged in a specific order. It is typically defined as an, where n represents the position in the sequence (e.g., n=1,2,3,).

2. **Check the general term**:
Identify the general formula for the sequence, an. This formula determines how each term in the sequence is calculated.

3. **Define the limit**:
The limit of a sequence, if it exists, is a number L such that as n approaches infinity (n), the terms of the sequence an get arbitrarily close to L. Mathematically, this is written as:
limnan=L.
If the terms do not approach a single value, the sequence is said to diverge.

4. **Simplify the expression for an**:
Simplify the general term if possible to make it easier to analyze. Factor out common terms, divide by the highest power of n, or use algebraic manipulations.

5. **Apply limit rules**:
Use standard limit rules and techniques, such as:
- **Divide by the highest power of n**: If the sequence involves fractions, divide numerator and denominator by the highest power of n to simplify the expression.
- **Recognize standard limits**: Some limits are well-known, such as:
limn1n=0,limnknp=0 (where k and p>0 are constants).
- **L'Hôpital's Rule**: If the sequence can be expressed as a function and is in an indeterminate form (e.g., 00 or ), L'Hôpital's rule may be applied.

6. **Conclude the result**:
After simplification, determine if the sequence converges (approaches a finite limit L) or diverges (does not settle to a specific value).

### Example: Find the limit of the sequence an=2n+3n+5.

1. The general term is an=2n+3n+5.
2. Simplify by dividing numerator and denominator by n (the highest power of n):
an=2nn+3nnn+5n=2+3n1+5n.
3. Take the limit as n:
- 3n0 and 5n0 as n grows large.
- Thus, an2+01+0=2.

4. Conclusion: The sequence converges, and its limit is L=2.

This approach can be adapted to different types of sequences by applying the appropriate simplifications and limit rules.

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