Find Limit of Xn as n Approaches Infinity

Thread starter dannysaf
Start date Sep 17, 2009
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Limit

In summary, the concept of a limit is a fundamental idea in calculus that describes the behavior of a mathematical function or sequence as the input value approaches a certain value. When a limit approaches infinity, it means that the input value is getting larger and larger without bound. The limit of a function or sequence is calculated by evaluating the function or sequence at values that are closer and closer to the desired input value. Finding the limit of a function or sequence is important because it helps us understand its behavior at a particular point and can be used in real-world applications such as physics, engineering, and economics.

Sep 17, 2009

dannysaf

Consider the sequence xn in which xn = 1/2(xn−1 + (3/xn-1) and x1 = a
(a not equals 0). Find lim n →∞ xn

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Sep 17, 2009

VeeEight

If you do not need to show that the limit actually exists, then you can go straight to finding it algebraically. Notice that the lim x_n is the same as lim x_(n-1) (as n tends to infinite). You can use this to find your limit.

1. What is the concept of a limit?

The concept of a limit is a fundamental idea in calculus that describes what happens to a mathematical function or sequence as the input value approaches a certain value. In other words, it defines the behavior of a function as the input value gets closer and closer to a particular value.

2. What does it mean for a limit to approach infinity?

When a limit approaches infinity, it means that the input value is getting larger and larger without bound. In other words, there is no specific value that the input is approaching, but rather it is getting infinitely bigger.

3. How is the limit of a function or sequence calculated?

The limit of a function or sequence is calculated by evaluating the function or sequence at values that are closer and closer to the desired input value. This is typically done using algebraic or analytical methods, such as substitution or L'Hôpital's rule.

4. What is the significance of finding the limit of a function or sequence?

Finding the limit of a function or sequence is important because it helps us understand the behavior of the function or sequence at a particular point. It can also be used to determine the behavior of the function or sequence as the input value approaches different values, such as infinity or negative infinity.

5. How is the limit of a function or sequence used in real-world applications?

The concept of a limit is used in many real-world applications, such as physics, engineering, and economics. For example, it can be used to determine the maximum capacity of a bridge or the rate at which a population is growing. It is also used in the development of mathematical models to predict and analyze real-world phenomena.