The concept of the limit of a function is essential to the study of calculus. It is used in defining some of the most important concepts in calculus—continuity, the derivative of a function, and the definite integral of a function.

The **limit** of a function *f*( *x*) describes the behavior of the function close to a particular *x* value. It does not necessarily give the value of the function at *x*. You write , which means that as *x* “approaches” *c*, the function *f*( *x*) “approaches” the real number *L* (see Figure 1).

**Figure 1 **The limit of f(x) as x approaches c.

In other words, as the independent variable *x* gets closer and closer to *c*, the function value *f*( *x*) gets closer to *L*. Note that this does not imply that *f*( *c*) = *L*; in fact, the function may not even exist at *c* (Figure 2) or may equal some value different than *L* at *c* (Figure 3).

**Figure 2*** f* ( *c*) does not exist, but does.

If the function does not approach a real number *L* as *x* approaches *c*, the limit does not exist; therefore, you write DNE (Does Not Exist). Many different situations could occur in determining that the limit of a function does not exist as *x* approaches some value.