Limit

For a real function $\displaystyle f(x)$ and some value $\displaystyle c$ in the domain of $\displaystyle f, \lim_{x\rightarrow c} f(x)$ (pronounced, "the limit of $f$ of $x$ as $x$ goes to $c$ ) equals $\displaystyle L$ iff for every $\displaystyle \epsilon > 0$ there exists a $\displaystyle \delta$ such that if $\displaystyle 0<|x-c|<\delta$ , then $\displaystyle |f(x)-L|< \epsilon$ .

Intuitive Meaning

The definition of a limit is a difficult thing to grasp, so many books give an intuitive definition first: a limit is the value to which the rest of the function grows closer. For example, $\displaystyle\lim_{x\rightarrow 2}x^2=4$ , because as the function $x$ grows arbitrarily close to 2 from either direction, the function $\displaystyle f(x)=x^2$ grows arbitrarily close to 4. In this case, the limit of the function is exactly equal to the value of the function. That is, $\displaystyle \lim_{x\rightarrow c} f(x) = f(c)$ . Unfortunately, this does not hold true in general. For example, consider the function $\displaystyle f(x)$ over the reals defined to be 0 if $\displaystyle x\neq 0$ and 1 if $\displaystyle x=0$ . Although the value of the function at 0 is 1, the limit $\displaystyle \lim_{x\rightarrow 0}f(x)$ is in fact zero. Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero, $\displaystyle f(x)$ will always be close to (in fact equal to) zero. Note that if our definition required only that $\displaystyle |x-c|<\delta$ , the limit of this function would not exist.

Other Matters

Limits do not always exist
Limits can be added, subtracted, and mulitplied
If a limit exists, it is unique

This article is a stub. Help us out by expanding it.

This article could use a lot of work -- there are formal errors (c doesn't have to be in the domain) and lots of omissions (limits from only one side, limits in a more general setting than the real line, continuity and its relation to limits, etc.)

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Limit

Intuitive Meaning

Other Matters