2009
10.31

The limit of a function f(x)

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The limit of \emph{f}(x) as x approaches a is the number L, written

\mathop {\lim }\limits_{x \to a} \emph{f}(x) = L


provided the \emph{f}(x) is arbitrarily close to L for all x sufficiently close to, but not equal to, a. A Limit is independent of the way (from the left or from the right) in which x approaches a. Here are some properties of limits:

  • \text{If }\emph{f}(x) = c \text{ is a constant function, then}\mathop {\lim }\limits_{x \to a} \emph{f}(x) = \mathop {\lim }\limits_{x \to a} c = c
  • \mathop {\lim }\limits_{x \to a} x^{n} = a^{n} \text{ for any positive integer } n
\text{If }\mathop {\lim }\limits_{x \to a} \emph{f}(x) = L_{1} \text{ and } \mathop {\lim }\limits_{x \to a} \emph{g}(x) = L_{2} \text{, where } L_{1} \text{ and } L_{2} \text{ are real numbers, then}


  • \mathop {\lim }\limits_{x \to a} [ \emph{f}(x) \pm \emph{g}(x) ] = \mathop {\lim }\limits_{x \to a} \emph{f}(x) \pm \mathop {\lim }\limits_{x \to a} \emph{g}(x) = L_{1} \pm L_{2}
  • \mathop {\lim }\limits_{x \to a} [ \emph{f}(x) \cdot \emph{g}(x) ] = \mathop {\lim }\limits_{x \to a} \emph{f}(x) \cdot \mathop {\lim }\limits_{x \to a} \emph{g}(x) = L_{1} \cdot L_{2}
  • \mathop {\lim }\limits_{x \to a} [ c\emph{f}(x) ] = c \cdot \mathop {\lim }\limits_{x \to a} \emph{f}(x) = cL_{2} \text{, where } c \text{ is a constant}
  • \text{If }\emph{f} \text{ is a polynomial function, then}\mathop {\lim }\limits_{x \to a} \emph{f}(x) = \mathop {\lim }\limits_{x \to a} \emph{f}(a)
  • \mathop {\lim }\limits_{x \to a} \frac{ \emph{f}(x) }{ \emph{g}(x) } = \frac{ \mathop {\lim }\limits_{x \to a} \emph{f}(x) }{ \mathop {\lim }\limits_{x \to a} \emph{g}(x) } = \frac{ L_{1} }{ L_{2} } \text{, if } { L_{2} } \neq 0
  • \mathop {\lim }\limits_{x \to a} \sqrt[n]{\emph{f}(x)} = \sqrt[n]{\mathop {\lim }\limits_{x \to a} \emph{f}(a)} = \sqrt[n]{L_{1}} *

* If n is even, L_{1} must be positive

Tip! Wolfram Alpha makes calculating limits really easy. Check this out.

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