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.
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.
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.
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.
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.
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.