Namespaces
Variants
Actions

Difference between revisions of "Indefinite limits and expressions, evaluations of"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
Methods for computing limits of functions given by formulas that cease to have a meaning when the limiting values of the argument are formally substituted in them, that is, go over into expressions like
 
Methods for computing limits of functions given by formulas that cease to have a meaning when the limiting values of the argument are formally substituted in them, that is, go over into expressions like
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i0505601.png" /></td> </tr></table>
+
$$\frac00,\frac\infty\infty,0\cdot\infty,\infty-\infty,0^0,\infty^0,1^\infty,$$
  
for which one cannot judge whether the required limits exist or not without saying anything about finding their values if they exist. The basic instrument of evaluating indeterminacies is Taylor's formula (cf. [[Taylor formula|Taylor formula]]), by means of which one singles out the principal part of a function. Thus, in the case of an indeterminacy of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i0505602.png" />, for which in order to find the limit
+
for which one cannot judge whether the required limits exist or not without saying anything about finding their values if they exist. The basic instrument of evaluating indeterminacies is Taylor's formula (cf. [[Taylor formula|Taylor formula]]), by means of which one singles out the principal part of a function. Thus, in the case of an indeterminacy of the type $0/0$, for which in order to find the limit
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i0505603.png" /></td> </tr></table>
+
$$\lim_{x\to x_0}\frac{f(x)}{g(x)},$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i0505604.png" /></td> </tr></table>
+
$$\lim_{x\to x_0}f(x)=\lim_{x\to x_0}g(x)=0,$$
  
one represents the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i0505605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i0505606.png" /> by Taylor's formulas in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i0505607.png" /> (if this is possible) up to the first non-zero term:
+
one represents the functions $f$ and $g$ by Taylor's formulas in a neighbourhood of $x_0$ (if this is possible) up to the first non-zero term:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i0505608.png" /></td> </tr></table>
+
$$f(x)=a(x-x_0)^n+o((x-x_0)^n),\quad a\neq0,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i0505609.png" /></td> </tr></table>
+
$$g(x)=b(x-x_0)^m+o((x-x_0)^m),\quad b\neq0;$$
  
 
as a result one finds that
 
as a result one finds that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056010.png" /></td> </tr></table>
+
$$\lim_{x\to x_0}\frac{f(x)}{g(x)}=\frac ab\lim_{x\to x_0}(x-x_0)^{n-m}=\begin{cases}0&\text{if}&n>m,\\\frac ab&\text{if}&n=m,\\\infty&\text{if}&n<m.\end{cases}$$
  
In the case of an indeterminacy of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056011.png" />, in order to find the limit
+
In the case of an indeterminacy of the type $\infty/\infty$, in order to find the limit
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056012.png" /></td> </tr></table>
+
$$\lim_{x\to x_0}\frac{f(x)}{g(x)},$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056013.png" /></td> </tr></table>
+
$$\lim_{x\to x_0}f(x)=\lim_{x\to x_0}g(x)=\infty,$$
  
 
one applies the transformation
 
one applies the transformation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056014.png" /></td> </tr></table>
+
$$\frac{f(x)}{g(x)}=\frac{\frac{1}{g(x)}}{\frac{1}{f(x)}},$$
  
which reduces the problem to the evaluation of an indeterminacy of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056015.png" />.
+
which reduces the problem to the evaluation of an indeterminacy of type $0/0$.
  
Indeterminacies of the types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056016.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056017.png" /> are also conveniently reduced to type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056018.png" /> by the following transformations:
+
Indeterminacies of the types $0\cdot\infty$ or $\infty-\infty$ are also conveniently reduced to type $0/0$ by the following transformations:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056019.png" /></td> </tr></table>
+
$$f(x)g(x)=\frac{f(x)}{\frac{1}{g(x)}}=\frac{g(x)}{\frac{1}{f(x)}},\quad f(x)-g(x)=\frac{\frac{1}{g(x)}-\frac{1}{f(x)}}{\frac{1}{f(x)}\frac{1}{g(x)}},$$
  
 
respectively.
 
respectively.
  
For evaluating indeterminacies of the types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056021.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056022.png" /> it is appropriate first to take the logarithm of the expressions whose limits are to be found.
+
For evaluating indeterminacies of the types $0^0$, $\infty^0$ or $1^\infty$ it is appropriate first to take the logarithm of the expressions whose limits are to be found.
  
Another general method for evaluating indeterminacies of the types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056023.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050560/i05056024.png" /> and those reducible to them is the [[L'Hospital rule|l'Hospital rule]].
+
Another general method for evaluating indeterminacies of the types $0/0$ or $\infty/\infty$ and those reducible to them is the [[L'Hospital rule|l'Hospital rule]].
  
  

Latest revision as of 18:01, 3 August 2014

Methods for computing limits of functions given by formulas that cease to have a meaning when the limiting values of the argument are formally substituted in them, that is, go over into expressions like

$$\frac00,\frac\infty\infty,0\cdot\infty,\infty-\infty,0^0,\infty^0,1^\infty,$$

for which one cannot judge whether the required limits exist or not without saying anything about finding their values if they exist. The basic instrument of evaluating indeterminacies is Taylor's formula (cf. Taylor formula), by means of which one singles out the principal part of a function. Thus, in the case of an indeterminacy of the type $0/0$, for which in order to find the limit

$$\lim_{x\to x_0}\frac{f(x)}{g(x)},$$

where

$$\lim_{x\to x_0}f(x)=\lim_{x\to x_0}g(x)=0,$$

one represents the functions $f$ and $g$ by Taylor's formulas in a neighbourhood of $x_0$ (if this is possible) up to the first non-zero term:

$$f(x)=a(x-x_0)^n+o((x-x_0)^n),\quad a\neq0,$$

$$g(x)=b(x-x_0)^m+o((x-x_0)^m),\quad b\neq0;$$

as a result one finds that

$$\lim_{x\to x_0}\frac{f(x)}{g(x)}=\frac ab\lim_{x\to x_0}(x-x_0)^{n-m}=\begin{cases}0&\text{if}&n>m,\\\frac ab&\text{if}&n=m,\\\infty&\text{if}&n<m.\end{cases}$$

In the case of an indeterminacy of the type $\infty/\infty$, in order to find the limit

$$\lim_{x\to x_0}\frac{f(x)}{g(x)},$$

where

$$\lim_{x\to x_0}f(x)=\lim_{x\to x_0}g(x)=\infty,$$

one applies the transformation

$$\frac{f(x)}{g(x)}=\frac{\frac{1}{g(x)}}{\frac{1}{f(x)}},$$

which reduces the problem to the evaluation of an indeterminacy of type $0/0$.

Indeterminacies of the types $0\cdot\infty$ or $\infty-\infty$ are also conveniently reduced to type $0/0$ by the following transformations:

$$f(x)g(x)=\frac{f(x)}{\frac{1}{g(x)}}=\frac{g(x)}{\frac{1}{f(x)}},\quad f(x)-g(x)=\frac{\frac{1}{g(x)}-\frac{1}{f(x)}}{\frac{1}{f(x)}\frac{1}{g(x)}},$$

respectively.

For evaluating indeterminacies of the types $0^0$, $\infty^0$ or $1^\infty$ it is appropriate first to take the logarithm of the expressions whose limits are to be found.

Another general method for evaluating indeterminacies of the types $0/0$ or $\infty/\infty$ and those reducible to them is the l'Hospital rule.


Comments

References

[a1] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Indefinite limits and expressions, evaluations of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Indefinite_limits_and_expressions,_evaluations_of&oldid=32712
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article