When we say "n approaches infinity" in the context of limits, it means that we are looking at the behavior of a function as the input value (n) gets larger and larger without bound. In other words, we are interested in what happens to the output of the function as the input becomes infinitely large.
To find the limit as n approaches infinity, we can use a few different methods. One approach is to substitute larger and larger values for n into the function and see if the output approaches a specific number. Another method is to use algebraic techniques such as factoring or simplifying to evaluate the limit. Additionally, we can use the rules of limits, such as the limit of a sum or product, to find the limit as n approaches infinity.
A finite limit is a limit where the output of the function approaches a specific number as the input approaches a certain value. This means that the function has a finite value at that point. On the other hand, an infinite limit is a limit where the output of the function approaches either positive or negative infinity as the input approaches a certain value. This means that the function does not have a finite value at that point.
Yes, the limit as n approaches infinity can be undefined. This can happen when the function does not approach a specific value or infinity as the input approaches infinity. In these cases, the limit is said to be "DNE" or "does not exist". It is important to note that a limit can be undefined even if the function is defined for all values of n.
Limits as n approaches infinity can be used in various real world applications, such as in population growth models, compound interest calculations, and analyzing the behavior of physical systems. For example, in population growth models, the limit as time (n) approaches infinity can help us predict the long-term growth rate of a population. In compound interest calculations, the limit as the number of compounding periods (n) approaches infinity can help us determine the total amount of interest earned over time. Overall, limits as n approaches infinity can provide valuable insights and predictions in many real world scenarios.