Limit at infinity refers to the mathematical concept of determining the behavior of a function as the input approaches positive or negative infinity. It is a fundamental principle in calculus that helps analyze the long-term trend or ultimate value of a function as the input values become larger and larger (or smaller and smaller).
Formally, the limit at infinity is denoted as lim f(x) as x approaches infinity (or negative infinity). It represents the value that the function, f(x), approaches as the input, x, grows without bound.
There are three possible outcomes for the limit at infinity. Firstly, the limit may not exist, meaning that the function does not approach a particular number as x becomes infinitely large or small. Secondly, the limit may be finite, indicating that the function approaches a specific value as x grows without bound. Lastly, the limit may be infinite, illustrating that the function grows without bound as x increases or decreases indefinitely.
The concept of a limit at infinity is useful in various areas of mathematics, including calculus, analysis, and differential equations. It helps determine asymptotic behavior, such as the presence of horizontal or slant asymptotes, as well as overall trends in functions over infinite ranges. Additionally, studying the limit at infinity provides insights into the concept of infinity and the behavior of functions at the extremes of their domain.
