In calculus, a limit is a value that a function approaches as the input (or independent variable) approaches a certain value. Limits are fundamental in calculus as they help us understand the behavior of functions near specific points and provide a foundation for both derivatives and integrals.
Limits can be defined for both continuous and discontinuous functions. In some cases, a limit may not exist, which indicates that the function’s behavior is not predictable or well-defined near that point.
To evaluate limits, mathematicians use various methods such as direct substitution, factorization, rationalization, and applying special limit properties. The concept of limits can be extended to include infinite limits and limits at infinity, which deal with the behavior of functions as they approach infinity or as their inputs approach infinity.
In addition to their importance in calculus, limits have applications in other areas of mathematics, such as real analysis and optimization problems.