Editing Limit of a function (section)

==Properties==
If a function {{mvar|f}} is real-valued, then the limit of {{mvar|f}} at {{mvar|p}} is {{mvar|L}} if and only if both the right-handed limit and left-handed limit of {{mvar|f}} at {{mvar|p}} exist and are equal to {{mvar|L}}.{{sfnp|Swokowski|1979|p=73}}

The function {{mvar|f}} is [[continuous function|continuous]] at {{mvar|p}} if and only if the limit of {{math|''f''(''x'')}} as {{mvar|x}} approaches {{mvar|p}} exists and is equal to {{math|''f''(''p'')}}.  If {{math|''f'' : ''M'' → ''N''}} is a function between metric spaces {{mvar|M}} and {{mvar|N}}, then it is equivalent that {{mvar|f}} transforms every sequence in {{mvar|M}} which converges towards {{mvar|p}} into a sequence in {{mvar|N}} which converges towards {{math|''f''(''p'')}}.

If {{mvar|N}} is a [[normed vector space]], then the limit operation is linear in the following sense: if the limit of {{math|''f''(''x'')}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|L}} and the limit of {{math|''g''(''x'')}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|P}}, then the limit of {{math|''f''(''x'') + g(''x'')}} as {{mvar|x}} approaches {{mvar|p}} is {{math|''L'' + ''P''}}. If  {{mvar|a}} is a scalar from the base [[field (mathematics)|field]], then the limit of {{math|''af''(''x'')}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|aL}}.

If {{mvar|f}} and {{mvar|g}} are real-valued (or complex-valued) functions, then taking the limit of an operation on {{math|''f''(''x'')}} and {{math|''g''(''x'')}} (e.g., {{math|''f'' + ''g''}}, {{math|''f'' &minus; ''g''}}, {{math|''f'' &times; ''g''}}, {{math|''f'' / ''g''}}, {{mvar|f{{sup| g}}}}) under certain conditions is compatible with the operation of limits of {{math|''f''(''x'')}} and {{math|''g''(''x'')}}. This fact is often called the '''algebraic limit theorem'''. The main condition needed to apply the following rules is that the limits on the right-hand sides of the equations exist (in other words, these limits are finite values including 0). Additionally, the identity for division requires that the denominator on the right-hand side is non-zero (division by 0 is not defined), and the identity for exponentiation requires that the base is positive, or zero while the exponent is positive (finite).

<math display=block>\begin{array}{lcl}
\displaystyle \lim_{x \to p} (f(x) + g(x)) & = & \displaystyle \lim_{x \to p} f(x) + \lim_{x \to p} g(x) \\
\displaystyle \lim_{x \to p} (f(x) - g(x)) & = & \displaystyle \lim_{x \to p} f(x) - \lim_{x \to p} g(x) \\
\displaystyle \lim_{x \to p} (f(x)\cdot g(x)) & = & \displaystyle \lim_{x \to p} f(x) \cdot \lim_{x \to p} g(x) \\
\displaystyle \lim_{x \to p} (f(x)/g(x)) & = & \displaystyle {\lim_{x \to p} f(x) / \lim_{x \to p} g(x)} \\
\displaystyle \lim_{x \to p} f(x)^{g(x)} & = & \displaystyle {\lim_{x \to p} f(x) ^ {\lim_{x \to p} g(x)}}
\end{array}</math>

These rules are also valid for one-sided limits, including when {{mvar|p}} is ∞ or −∞. In each rule above, when one of the limits on the right is ∞ or −∞, the limit on the left may sometimes still be determined by the following rules.

<math display=block>\begin{array}{rcl}
  q + \infty & = & \infty \text{ if } q \neq -\infty \\[8pt]
  q \times \infty & = & \begin{cases} 
    \infty & \text{if } q > 0 \\ 
    -\infty & \text{if } q < 0 
    \end{cases} \\[6pt]
  \displaystyle \frac q \infty & = & 0 \text{ if } q \neq \infty \text{ and } q \neq -\infty \\[6pt]
  \infty^q & = & \begin{cases} 
    0 & \text{if } q < 0 \\
    \infty & \text{if } q > 0 
    \end{cases} \\[4pt]
  q^\infty & = & \begin{cases}
    0 & \text{if } 0 < q < 1 \\
    \infty & \text{if }  q > 1 
    \end{cases} \\[4pt]
  q^{-\infty} & = & \begin{cases} 
    \infty & \text{if } 0 < q < 1 \\
    0 & \text{if } q > 1
    \end{cases}
\end{array}</math>

(see also [[Extended real number line]]).

In other cases the limit on the left may still exist, although the right-hand side, called an ''[[indeterminate form]]'', does not allow one to determine the result. This depends on the functions {{mvar|f}} and {{mvar|g}}. These indeterminate forms are:

<math display=block>\begin{array}{cc}
\displaystyle \frac{0}{0} & \displaystyle \frac{\pm \infty}{\pm \infty} \\[6pt]
0 \times \pm \infty & \infty + -\infty \\[8pt]
\qquad 0^0 \qquad & \qquad \infty^0 \qquad \\[8pt]
1^{\pm \infty}
\end{array}</math>

See further [[#L'Hôpital's rule|L'Hôpital's rule]] below and [[Indeterminate form]].

===Limits of compositions of functions===
In general, from knowing that <math>\lim_{y \to b} f(y) = c</math> and <math>\lim_{x \to a} g(x) = b,</math> it does ''not'' follow that <math>\lim_{x \to a} f(g(x)) = c.</math> 
However, this "chain rule" does hold if one of the following ''additional'' conditions holds:
*{{math|1=''f''(''b'') = ''c''}} (that is, {{mvar|f}} is continuous at  {{mvar|b}}), or
*{{mvar|g}} does not take the value  {{mvar|b}} near  {{mvar|a}} (that is, there exists a {{math|''δ'' > 0}} such that if {{math|0 < {{abs|''x'' &minus; ''a''}} < ''δ''}} then {{math|{{abs|''g''(''x'') &minus; ''b''}} > 0}}).

As an example of this phenomenon, consider the following function that violates both additional restrictions:

<math display=block>f(x) = g(x) = \begin{cases}
  0 & \text{if } x\neq 0 \\ 
  1 & \text{if } x=0 
\end{cases}</math>

Since the value at {{math|''f''(0)}} is a [[removable discontinuity]],
<math display=block>\lim_{x \to a} f(x) = 0</math> for all {{mvar|a}}.
Thus, the naïve chain rule would suggest that the limit of {{math|''f''(''f''(''x''))}} is 0.  However, it is the case that
<math display=block>f(f(x))=\begin{cases}
  1 & \text{if } x\neq 0 \\ 
  0 & \text{if } x = 0 
\end{cases}</math>
and so
<math display=block>\lim_{x \to a} f(f(x)) = 1</math> for all {{mvar|a}}.

===Limits of special interest===
{{main|List of limits}}

====Rational functions====

For {{mvar|n}} a nonnegative integer and constants <math>a_1, a_2, a_3,\ldots, a_n</math> and <math>b_1, b_2, b_3,\ldots, b_n,</math>

<math display=block>\lim_{x \to \infty} \frac{a_1 x^n + a_2 x^{n-1} + a_3 x^{n-2} + \dots + a_n}{b_1 x^n + b_2 x^{n-1} + b_3 x^{n-2} + \dots + b_n} = \frac{a_1}{b_1}</math>

This can be proven by dividing both the numerator and denominator by {{mvar|x{{sup|n}}}}. If the numerator is a polynomial of higher degree, the limit does not exist. If the denominator is of higher degree, the limit is 0.

====Trigonometric functions====

<math display=block>\begin{array}{lcl}
\displaystyle \lim_{x \to 0} \frac{\sin x}{x} & = & 1 \\[4pt]
\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} & = & 0
\end{array}</math>

====Exponential functions====

<math display=block>\begin{array}{lcl}
  \displaystyle \lim_{x \to 0} (1+x)^{\frac{1}{x}} & = & \displaystyle \lim_{r \to \infty} \left(1+\frac{1}{r}\right)^r = e \\[4pt]
  \displaystyle \lim_{x \to 0} \frac{e^{x}-1}{x} & = & 1 \\[4pt]
  \displaystyle \lim_{x \to 0} \frac{e^{ax}-1}{bx} & = & \displaystyle \frac{a}{b} \\[4pt]
  \displaystyle \lim_{x \to 0} \frac{c^{ax}-1}{bx} & = & \displaystyle \frac{a}{b}\ln c \\[4pt]
  \displaystyle \lim_{x \to 0^+} x^x & = & 1
\end{array}</math>

====Logarithmic functions====

<math display=block>\begin{array}{lcl}
  \displaystyle \lim_{x \to 0} \frac{\ln(1+x)}{x} & = & 1 \\[4pt]
  \displaystyle \lim_{x \to 0} \frac{\ln(1+ax)}{bx} & = & \displaystyle \frac{a}{b} \\[4pt]
  \displaystyle \lim_{x \to 0} \frac{\log_c(1+ax)}{bx} & = & \displaystyle \frac{a}{b\ln c}
\end{array}</math>

===L'Hôpital's rule===
{{Main|l'Hôpital's rule}}
This rule uses [[derivative]]s to find limits of [[indeterminate forms]] {{math|0/0}} or {{math|±∞/∞}}, and only applies to such cases.  Other indeterminate forms may be manipulated into this form.  Given two functions {{math|''f''(''x'')}} and {{math|''g''(''x'')}}, defined over an [[open interval]] {{mvar|I}} containing the desired limit point {{mvar|c}}, then if:
# <math>\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0,</math>  or <math>\lim_{x \to c}f(x)=\pm\lim_{x \to c}g(x) = \pm\infty,</math> and
# <math>f</math> and <math>g</math> are differentiable over <math>I \setminus \{c\},</math> and
# <math>g'(x)\neq 0</math> for all <math> x \in I \setminus \{c\},</math> and
# <math>\lim_{x\to c}\tfrac{f'(x)}{g'(x)}</math>  exists,
then:
<math display=block>\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}.</math>

Normally, the first condition is the most important one.

For example:
<math>\lim_{x \to 0} \frac{\sin (2x)}{\sin (3x)} =
\lim_{x \to 0} \frac{2 \cos (2x)}{3 \cos (3x)} =
\frac{2 \sdot 1}{3 \sdot 1} =
\frac{2}{3}.</math>

===Summations and integrals===
Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.

A short way to write the limit <math>\lim_{n \to \infty} \sum_{i=s}^n f(i) </math>
is <math>\sum_{i=s}^\infty f(i).</math> An important example of limits of sums such as these are [[Series (mathematics)|series]].

A short way to write the limit <math>\lim_{x \to \infty} \int_a^x f(t) \; dt </math>
is <math>\int_a^\infty f(t) \; dt.</math>

A short way to write the limit <math>\lim_{x \to -\infty} \int_x^b f(t) \; dt </math>
is <math>\int_{-\infty}^b f(t) \; dt.</math>