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 , where represents the position in the sequence (e.g., ).
2. **Check the general term**:
Identify the general formula for the sequence, . 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 such that as approaches infinity (), the terms of the sequence get arbitrarily close to . Mathematically, this is written as:
If the terms do not approach a single value, the sequence is said to diverge.
4. **Simplify the expression for **:
Simplify the general term if possible to make it easier to analyze. Factor out common terms, divide by the highest power of , or use algebraic manipulations.
5. **Apply limit rules**:
Use standard limit rules and techniques, such as:
- **Divide by the highest power of **: If the sequence involves fractions, divide numerator and denominator by the highest power of to simplify the expression.
- **Recognize standard limits**: Some limits are well-known, such as:
- **L'Hôpital's Rule**: If the sequence can be expressed as a function and is in an indeterminate form (e.g., or ), L'Hôpital's rule may be applied.
6. **Conclude the result**:
After simplification, determine if the sequence converges (approaches a finite limit ) or diverges (does not settle to a specific value).
### Example: Find the limit of the sequence .
1. The general term is .
2. Simplify by dividing numerator and denominator by (the highest power of ):
3. Take the limit as :
- and as grows large.
- Thus, .
4. Conclusion: The sequence converges, and its limit is .
This approach can be adapted to different types of sequences by applying the appropriate simplifications and limit rules.