When a function has a limit at infinity, it means that as the input values of the function approach infinity, the output values of the function also approach a specific number. This number is known as the limit. It represents the value that the function gets closer and closer to, but never reaches, as the input value increases without bound.
To determine the limit of a function at infinity, you need to evaluate the function at extremely large input values, such as 100, 1000, or even larger. If the output values of the function appear to be approaching a specific number, then that number is the limit. You can also use algebraic techniques, such as factoring and simplifying, to determine the limit at infinity.
No, a function can only have one limit at infinity. This is because the limit represents the unique value that the function approaches as the input values get larger and larger. If the function has multiple limits at infinity, it would mean that the function is approaching different values simultaneously, which is not possible.
The behavior of a function at infinity is directly related to its limit. If the function approaches a specific number as the input values increase without bound, then the limit will also be that number. However, if the function oscillates or approaches different values, then the limit at infinity does not exist.
Yes, a function can have a limit at infinity even if it is not defined at infinity. This is because the limit at infinity only considers the behavior of the function as the input values approach infinity, but it does not necessarily require the function to be defined at infinity. For example, the function f(x) = 1/x has a limit of 0 at infinity, even though it is not defined at x = 0.