A limit in mathematics is a fundamental concept used to describe the behavior of a function as its input approaches a certain value. It represents the value that a function approaches as its input gets closer and closer to a specified value. In other words, it is the value that a function "approaches" but may never actually reach.
To compute a limit, you must first determine the behavior of the function as it approaches the specified value. This can be done by evaluating the function at values that are closer and closer to the specified value. If the function approaches a specific value, then that value is the limit. If the function approaches different values from the left and right sides, or if it approaches infinity, then the limit does not exist.
A one-sided limit only considers the behavior of a function as it approaches the specified value from one side (either the left or the right). This means that the limit only exists if the function approaches the same value from both sides. A two-sided limit, on the other hand, considers the behavior of the function as it approaches the specified value from both the left and right sides.
The most common methods for computing limits include direct substitution, factoring, and rationalizing the numerator or denominator. For more complicated functions, you may need to use techniques such as L'Hôpital's rule, trigonometric identities, or algebraic manipulation.
Limits are used in real-life applications to model and predict the behavior of various systems and processes. For example, they can be used to determine the maximum safe speed of a vehicle on a curved road or the maximum safe dosage of a medication. They are also used in various fields of science, such as physics, chemistry, and biology, to describe and analyze natural phenomena.