Indetermination of the form ∞/∞

In Analysis, the calculation of limit leads sometimes to the following situation: in a Quotient, the Numerator and the Dénominateur have both for limit the infinite one. In this case, no operational rule on the limits applies, one says that one deals with indetermination of the form $\ scriptstyle \ frac \ infty \ infty$ .

For example:

$\ lim_ {X \ to + \ infty} \ frac {e^x} {X}$

To raise the indetermination, there exist many processes, algebraic (Factorization) or analytical (use of the Dérivée, Théorème of the gendarmes or Développement limited).

Example

To raise the unspecified form above, one can for example pose a function such as:

\ operatorname {\ varphi} {(X)} =e^x-x

By deriving it, one obtains:

\ operatorname {\ varphi^ \ premium} {(X)} =e^x-1

The table of the variations of this function teaches us that it is decreasing on

] - \ infty; 0]

and increasing on

gold ~ \ operatorname {\ varphi} {(0)} =1, therefore:
: \ operatorname {\ varphi} {(X)} \ Ge 0

e^x - X \ Ge 0

e^x \ Ge x

One can thus apply the theorem of the gendarmes:

\ left.

\begin{matrix} \ displaystyle {\ frac {e^x} {X} \ Ge X} \ \ \ lim_ {X \ to + \ infty} x=+ \ infty \end{matrix} \ right \} \ Rightarrow \ lim_ {X \ to + \ infty} \ frac {e^x} {X} =+ \ infty

Internal bonds

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