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[parent] improper limits (Definition)

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

$\displaystyle \lim_{x \to 0}\frac{1}{x^2} = \infty.$
Such “limits” are actually extensions 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$.

$\displaystyle \lim_{x \to x_0}f(x) = \infty$
iff for every real number $ M$ there exists a number $ \delta_M$ such that
$\displaystyle f(x) > M$
as soon as
$\displaystyle 0 < \vert x-x_0\vert < \delta_M.$
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$.

Note 1. If $ \lim_{x \to x_0}f(x) = \infty$ and $ \lim_{x \to x_0}g(x) = a > 0$, then we have

$\displaystyle \lim_{x \to x_0}f(x)g(x) = \infty.$
Hence we can say that $ \infty\cdot a = \infty$ when $ a > 0$. There are some other comparable “mnemonics of infinite” (cf. the extended real numbers):
$\displaystyle \infty\cdot a = -\infty \quad(a < 0)$
$\displaystyle \pm\infty+a = \pm\infty$
$\displaystyle \frac{a}{\pm\infty} = 0$
$\displaystyle \infty+\infty = \infty$
$\displaystyle \infty\cdot\infty = \infty$
$\displaystyle -\infty\cdot\infty = -\infty$

On the contrary, there exist no mnemonics for the cases

$\displaystyle \infty\cdot0,\,\, \infty-\infty,\,\, \frac{\infty}{\infty},\,\, \frac{0}{0},\,\, 0^0,\,\, \infty^0,\,\, 1^\infty;$
they are indefinite and depend on the instance (cf. the indeterminate form).

Note 2. In the complex plane, the expression

$\displaystyle \lim_{z \to z_0}f(z) = \infty$
means that $ \displaystyle \lim_{z \to z_0}\vert f(z)\vert = \infty$.



"improper limits" is owned by pahio.
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See Also: l'Hôpital's rule, extended real numbers, limit rules of functions, integrating $\tan x$ over $[0,\frac{\pi}{2}]$, indeterminate form, example of jump discontinuity, list of common limits

Other names:  infinite limits
Also defines:  mnemonic of infinite

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Cross-references: complex plane, indeterminate form, mnemonics, extended real numbers, similar, number, real number, iff, point, neighbourhood, real function, limit, expressions, Calculus
There are 3 references to this entry.

This is version 20 of improper limits, born on 2004-10-03, modified 2008-01-14.
Object id is 6283, canonical name is ImproperLimits.
Accessed 4410 times total.

Classification:
AMS MSC: 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
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