Can you think of a basic theorem of limits that would lead to a one line proof?Karl Porter said:Homework Statement:: I was curious, if I have a sequence that has a limit of 1
Relevant Equations:: Lim an=1 as n tends to inf
Lim of an^2=1 as n tends to inf
Does the square of the sequence also have a limit of 1. Does the square root also equal 1? I've been trying to find some counterexamples but I think the limit doesn't change under these operations?
A limit of a sequence is the value that the terms of a sequence approach as the index approaches infinity.
To find the limit of a sequence, you can use various techniques such as the squeeze theorem, the ratio test, or the root test. These methods involve evaluating the behavior of the sequence as the index approaches infinity.
In general, taking the square or square root of a sequence does not change its limit. However, there are certain cases where the limit may change, such as when the sequence alternates between positive and negative values.
Yes, the limit of a sequence can be undefined if the terms of the sequence do not approach a specific value as the index approaches infinity. This can happen if the sequence oscillates or diverges.
The limit of a sequence is related to continuity because a function is continuous at a point if and only if the limit of its sequence of values at that point is equal to the function's value at that point. In other words, the limit of a sequence can help determine the continuity of a function.