Limite cu radical

Licuricy Sam Mar 17, 2012 5:02 pm

Cum as putea rezolva acesta limita: $lim x\rightarrow 1 \frac{1-\sqrt{x}}{1-x^2}=\frac{1-1}{1-1}=\frac{0}{0}$
cum trebuie sa o rezolv mai departe ?

**Admin** Sam Mar 17, 2012 7:38 pm

Fiind nedeterminare, vom apela la următorul "trick" matematic :
$\begin{array}{l} \mathop {{\rm{lim}}}\limits_{x \to 1} \frac{{1 - \sqrt x }}{{1 - {x^2}}} = \mathop {{\rm{lim}}}\limits_{x \to 1} \frac{{1 - \sqrt x }}{{\left( {1 - x} \right)\left( {1 + x} \right)}} = \mathop {{\rm{lim}}}\limits_{x \to 1} \frac{{1 - \sqrt x }}{{\left[ {1 - {{\left( {\sqrt x } \right)}^2}} \right]\left( {1 + x} \right)}}\\ = \mathop {{\rm{lim}}}\limits_{x \to 1} \frac{{1 - \sqrt x }}{{\left( {1 - \sqrt x } \right)\left( {1 + \sqrt x } \right)\left( {1 + x} \right)}} = \mathop {{\rm{lim}}}\limits_{x \to 1} \frac{1}{{\left( {1 + \sqrt x } \right)\left( {1 + x} \right)}} = \frac{1}{4} \end{array}$

Limite cu radical

Limite cu radical

Re: Limite cu radical