# Defining a Limit

A limit of a function is the value that function approaches as the independent variable of the function approaches a given value. The equation *f* (*x*) = *t* is equivalent to the statement “The limit of *f* as *x* goes to *c* is *t* .” Another way to phrase this equation is “As *x* approaches *c* , the value of *f* gets arbitrarily close to *t* .” This is the essential concept of a limit.

Here are some properties of limits.

x = a |

k = k, where k is a constant. |

x ^{b} = a ^{b} |

=if a≥ 0 |

Here are some properties of operations with limits. Let *f* (*x*) = *C* , and *g*(*x*) = *D* .

kf (x) = kC, where k is a constant. |

(f (x)±g(x)) = C±D |

f (x)×g(x) = C×D |

= , if D≠ 0 |

[f (x)]^{n} = C ^{n} |

The more formal definition of a limit is the following. *f* (*x*) = *A* if and only if for any positive number *ε* , there exists another positive number *δ* , such that if 0 < |*x* – *a*| < *ε* , then | *f* (*x*) – *A*| < *δ* . This definition basically states that if *A* is the limit of *f*as *x* approaches *a* , then any time *f* (*x*) is within *ε* units of a value *A* , another interval(*x* – *δ*, *x* + *δ*) exists such that all values of *f* (*x*) between (*x* – *δ*) and (*x* + *δ*) lie within the bounds (*A* – *ε*, *A* + *ε*) . A simpler way of saying it is this: if you choose an *x* -value*x* _{1} which is very close to *x* = *a* , there always exists another *x* -value *x* _{0} closer to *a*such that *f* (*x* _{0}) is closer to *f* (*a*) than *f* (*x* _{1}) .

A limit of a function can also be taken “from the left” and “from the right.” These are called one-sided limits. The equation *x*âÜ’*a* ^{–}]*f* (*x*) = *A* reads “The limit of *f* (*x*)as *x* approaches *a* from the left is *A* .” “From the left” means from values less than *a* — left refers to the left side of the graph of *f* . The equation *x*âÜ’*a* ^{+}]*f* (*x*) = *A*means that the limit is found by calculating values of *x* that approach *a* which are greater than *a* , or to the right of *a* in the graph of *f* .

There are a few cases in which a limit of a function *f* at a given *x* -value *a* does not exist. They are as follows: 1) If *x*âÜ’*a* ^{–}]*f* (*x*)≠ *x*âÜ’*a* ^{+}]*f* (*x*) . 2) If *f* (*x*)increases of decreases without bound as *x* approaches *a* . 3) If *f* oscillates (switches back and forth) between fixed values as *x* approaches *a* . In these situations, the limit of *f* (*x*) at *x* = *a* does not exist.

One of the most important things to remember about limits is this: *f* (*x*) is independent of *f* (*a*) . All that matters is the behavior of the function at the *x* -values *near* *a* , not at *a* . It is not uncommon for a function have a limit at an *x* -value for which the function is undefined.