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Intuitively, the **limit** of *f* (*x*) as *x* approaches *c* is the *value*
that *f* (*x*) approaches as *x* approaches *c*. For example, the limit of *f* (*x*) = *x*^{2} + 2
as *x* approaches 2 is 6:

Figure %: The Limit of *f* (*x*) = *x*^{2} + 2 as *x* approaches 2

As *x* gets closer and closer to 2, *f* (*x*) gets closer and closer to 6. In mathematical
notation, we can represent this as

f (x) = 6 or x^{2}+2 = 6 |

Note that we've only been talking about what happens to *f* (*x*) as *x**approaches**c*, and not about what happens when *x**equals**c*. The
truth is that when we're looking for limits, we don't care what happens to *f* (*x*) when
*x* actually equals *c* -- we're only concerned with its behavior as *x* gets closer and
closer to *c*. Consider the following piecewise-defined function:

f (x) = |

Note that this function looks just like the function *f* (*x*) = *x*^{2} + 2, except that *f* (2) = 9
instead of 6. What happens when we try to find

f (x) ? |

Figure %: The Limit of *f* (*x*) as *x* approaches 2

We see that the limit is again 6. Once again, this is because * the limit doesn't care
what happens when x = c!* As long as two functions approach the same value as

The standard limit that we've been talking about is a **two-sided limit**. It is
considered two-sided because we get the same value for the limit whether we let *x*
approach *c* "from the left" (i.e. from values of *x* less than *c*)

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