# improper limits

In calculus there is often used such expressions as “the limit of a function is infinite”, and one may write for instance that

 $\lim_{x\to 0}\frac{1}{x^{2}}\;=\;\infty.$

Such “limits” are actually of the limit notion, and can be defined exactly.  They are called improper limits.

Definition.  Let the real function $f$ be defined in a neighbourhood of the point $x_{0}$.

 $\lim_{x\to x_{0}}f(x)\;=\;\infty$

iff for every real number $M$ there exists a number $\delta_{M}$ such that

 $f(x)\;>\;M$

as soon as

 $0\;<\;|x\!-\!x_{0}|\;<\;\delta_{M}.$

In a similar way we can define the improper limit $-\infty$ of a real function.  The definition may be extended also to the cases  $x\to\pm\infty$, when one speaks of limits at infinity.

Note 1.  If  $\lim_{x\to x_{0}}f(x)\,=\,\infty$  and  $\lim_{x\to x_{0}}g(x)\,=\,a>0$,  then we have

 $\lim_{x\to x_{0}}f(x)g(x)\;=\;\infty.$

Hence we can say that  $\infty\cdot a=\infty$  when  $a>0$.  There are some other “mnemonics of infinite” (cf. the extended real numbers):

 $\infty\cdot a\;=\;-\infty\qquad(a\;<\;0)$
 $\pm\infty+a\;=\;\pm\infty$
 $\frac{a}{\pm\infty}\;=\;0$
 $\infty+\infty\;=\;\infty$
 $\infty\cdot\infty\;=\;\infty$
 $-\infty\cdot\infty\;=\;-\infty$

On the contrary, there exist no mnemonics for the cases

 $\infty\cdot 0,\,\,\infty-\infty,\,\,\frac{\infty}{\infty},\,\,\frac{0}{0},\,\,% 0^{0},\,\,\infty^{0},\,\,1^{\infty};$

they are and depend on the instance (cf. the indeterminate form).

Note 2.  In the complex plane, the expression

 $\lim_{z\to z_{0}}f(z)\;=\;\infty$

means that  $\displaystyle\lim_{z\to z_{0}}|f(z)|\,=\,\infty$.

 Title improper limits Canonical name ImproperLimits Date of creation 2013-03-22 14:40:45 Last modified on 2013-03-22 14:40:45 Owner pahio (2872) Last modified by pahio (2872) Numerical id 24 Author pahio (2872) Entry type Definition Classification msc 26A06 Synonym infinite limits Synonym improper limit Related topic LHpitalsRule Related topic ExtendedRealNumbers Related topic LimitRulesOfFunctions Related topic IntegratingTanXOver0fracpi2 Related topic IndeterminateForm Related topic ExampleOfJumpDiscontinuity Related topic ListOfCommonLimits Related topic LimitsOfNaturalLogarithm Related topic SecondDerivativeAsSimpleLimit Related topic AngleBetweenTwoLines Defines limit at infinity Defines mnemonic of infinite