Prove that lim gn for n→∞ exists, and find it.

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In summary, a limit exists when the output of a function approaches a specific value as the input approaches a certain point. It is important to prove that a limit exists in order to understand the behavior of a function and make predictions about its output. This can be done by showing that the function approaches the same value from both sides of the input point, using techniques such as the epsilon-delta definition, the squeeze theorem, or the limit laws. Notations such as "lim gn for n→∞" indicate the long-term behavior of a function as the input approaches larger values. Mathematical methods, such as the epsilon-delta definition, the squeeze theorem, and the limit laws, can be used to find the limit by manipulating the function algebra
  • #1
dannysaf
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Suppose g1 , g2 ,... are any numbers that satisfy the inequalities
0 < gn < 1 and (1 − gn )gn+1 > 1/4 for all n.

Prove that lim gn for n→∞ exists, and find it.


I need well substantiated answer! Thanks.
 
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  • #2
dannysaf said:
Suppose g1 , g2 ,... are any numbers that satisfy the inequalities
0 < gn < 1 and (1 − gn )gn+1 > 1/4 for all n.

Prove that lim gn for n→∞ exists, and find it.


I need well substantiated answer! Thanks.

Is this a homework problem? I'll give you a hint. Show:

[tex]\frac 1 {4(1-x)} \ge x[/tex]

on the interval, consequently showing your sequence gn is increasing. Then...
 

1. What does it mean for a limit to exist?

For a limit to exist, the function must approach a specific value as the input approaches a certain point. In other words, the output of the function must get closer and closer to a single value as the input gets closer and closer to a specific value.

2. Why is it important to prove that a limit exists?

Proving that a limit exists is important because it allows us to determine the behavior of a function as the input approaches a certain point. This helps us understand the overall behavior of the function and make predictions about its output.

3. How do you prove that a limit exists?

To prove that a limit exists, you must show that the function approaches the same value from both sides of the input point. This can be done by using the epsilon-delta definition of a limit or by using other techniques such as the squeeze theorem or the limit laws.

4. What is the significance of the notation "lim gn for n→∞"?

This notation indicates that we are looking at the behavior of the function as the value n approaches infinity. In other words, we are interested in the long-term behavior of the function as the input gets larger and larger.

5. How do you find the limit using mathematical methods?

There are several mathematical methods that can be used to find a limit, including the epsilon-delta definition, the squeeze theorem, and the limit laws. These methods involve manipulating the function algebraically or using properties of limits to determine the value that the function approaches as the input approaches a specific point.

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