# Limits and Continuity

The concept of the limit is one of the most crucial things to understand in order to prepare for calculus. A limit is a number that a function approaches as the independent variable of the function approaches a given value. For example, given the function *f* (*x*) = 3*x* , you could say, “The limit of *f* (*x*) as *x* approaches 2 is 6 .” Symbolically, this is written *f* (*x*) = 6 . In the following sections, we will more carefully define a limit, as well as give examples of limits of functions to help clarify the concept.

Continuity is another far-reaching concept in calculus. A function can either be continuous or discontinuous. One easy way to test for the continuity of a function is to see whether the graph of a function can be traced with a pen without lifting the pen from the paper. For the math that we are doing in precalculus and calculus, a conceptual definition of continuity like this one is probably sufficient, but for higher math, a more technical definition is needed. Using limits, we’ll learn a better and far more precise way of defining continuity as well. With an understanding of the concepts of limits and continuity, you are ready for calculus.

**Continuity**– A function is continuous at a point if the three following conditions are met: 1)

*f*(

*a*) is defined. 2)

*f*(

*x*) exists. 3)

*f*(

*x*) =

*f*(

*a*) . A conceptual way to describe continuity is this: A function is continuous if its graph can be traced with a pen without lifting the pen from the page.

**Infinite Discontinuity**– A category of discontinuity in which a vertical asymptote exists at

*x*=

*a*and

*f*(

*a*) is undefined.

**Jump Discontinuity**– A category of discontinuity in which

*f*(

*x*)≠

*f*(

*x*) , but both of these limits exist and are finite.

**Limit**– The value

*A*to which a function

*f*(

*x*) gets arbitrarily close as the value of the independent variable

*x*gets arbitrarily close to a given value

*a*. Such a limit is symbolized this way:

*lim*

_{xâÜ’a}

*f*(

*x*) =

*A*.

**One-Sided Limit**– A limit based entirely on the values of a function taken at an

*x*-value slightly greater than or less than a given value. Whereas a two-sided limit

*f*(

*x*) takes into account the values of

*x*near

*a*which are both greater than and less than

*a*, a one-sided limit from the left

*f*(

*x*) or from the right

*f*(

*x*) takes into account only values of

*x*less than

*a*, or greater than

*a*, respectively.

**Point Discontinuity**– A category of discontinuity in which a function has a well-defined two-sided limit at

*x*=

*a*, but either

*f*(

*x*) is not defined at

*a*or its value at

*a*is not equal to this limit.