A limit evaluation is a mathematical process used to find the value of a function at a specific point or as the input approaches a certain value. It is commonly used in calculus to determine the behavior of a function as the input gets closer and closer to a particular value.
Evaluating a limit involves plugging in the given value for the input and simplifying the resulting expression. If the resulting expression is undefined, further algebraic manipulations or other methods such as L'Hôpital's rule may be used to evaluate the limit.
Limits do not exist if the function approaches different values from the left and right sides of the given input, or if the function approaches positive or negative infinity at the given input. Additionally, limits do not exist at points where the function has a hole or a vertical asymptote.
A one-sided limit only considers the behavior of the function as the input approaches the given value from one side, either the left or the right. A two-sided limit, on the other hand, considers the behavior of the function as the input approaches the given value from both the left and right sides.
Limits are important in calculus because they allow us to study the behavior of functions in a more precise way. They are used to calculate derivatives, find maximum and minimum values, and determine the convergence or divergence of infinite series. They also provide a foundation for more advanced concepts such as continuity and differentiability.