dannysaf said:I need well substantiated answer! Thanks.
A limit in calculus is the value that a function approaches as its input (usually denoted as x) approaches a certain value. It is a fundamental concept in calculus and is used to analyze the behavior of functions as they approach specific values.
To prove that a limit exists, you need to show that the function approaches a single value as the input approaches a specific value. This can be done by using the epsilon-delta definition of a limit, where you can manipulate inequalities to show that the distance between the function and the limit value approaches zero as the input approaches the specific value.
If a limit exists, it means that the function approaches a single value as the input approaches a specific value. In other words, the function has a well-defined value at that specific input, even if it is not defined at that point.
Proving that a limit exists is important because it allows us to evaluate the behavior of functions at specific points, even if the function is not defined at that point. It also helps us understand the continuity of a function and its behavior near certain points.
To find the limit of a function as n approaches infinity, you can use the following steps:
1. Rewrite the function in a way that allows you to evaluate it as n approaches infinity (e.g. by factoring or simplifying).
2. Plug in infinity for n and evaluate the function.
3. If the limit is an indeterminate form (e.g. 0/0 or ∞/∞), you can use techniques such as L'Hopital's rule or algebraic manipulation to evaluate the limit.
4. If the limit is a defined value, then that is the limit of the function. Otherwise, the limit does not exist.