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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.

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