Regulile lui l`Hospital

Licuricy Vin Mai 25, 2012 8:15 pm

Sa se calculeze:
$limx\rightarrow 1,x>1 \frac{nx^{n+2}-\left ( n+1\right )x^{n+1}+x}{\left ( x-1\right )^2}$

**Admin** Vin Mai 25, 2012 8:50 pm

$\begin{array}{l} \mathop {{\rm{lim}}}\limits_{x \to 1} \frac{{n{x^{n + 2}} - \left( {n + 1} \right){x^{n + 1}} + x}}{{{{\left( {x - 1} \right)}^2}}} = \mathop {{\rm{lim}}}\limits_{x \to 1} \frac{{\left[ {n{x^{n + 2}} - \left( {n + 1} \right){x^{n + 1}} + x} \right]'}}{{\left[ {{{\left( {x - 1} \right)}^2}} \right]'}}\\ \left[ {n{x^{n + 2}} - \left( {n + 1} \right){x^{n + 1}} + x} \right]' = n\left( {n + 2} \right){x^{n + 1}} - {\left( {n + 1} \right)^2}{x^n} + 1\\ \left[ {{{\left( {x - 1} \right)}^2}} \right]' = 2\left( {x - 1} \right) \end{array}$
$L = \mathop {{\rm{lim}}}\limits_{x \to 1} \frac{{n\left( {n + 2} \right){x^{n + 1}} - {{\left( {n + 1} \right)}^2}{x^n} + 1}}{{2\left( {x - 1} \right)}}\,{\,^{^{^{^{^{^{0/0}}}}}}}$
Vom aplica din nou regula lui l'Hospital:
${L = \mathop {{\rm{lim}}}\limits_{x \to 1} \frac{{n\left( {n + 2} \right){x^{n + 1}} - {{\left( {n + 1} \right)}^2}{x^n} + 1}}{{2\left( {x - 1} \right)}}{\mkern 1mu} = \frac{1}{2} \times \mathop {{\rm{lim}}}\limits_{x \to 1} \frac{{{{\left[ {n\left( {n + 2} \right){x^{n + 1}} - {{\left( {n + 1} \right)}^2}{x^n} + 1} \right]}^\prime }}}{{{{\left[ {\left( {x - 1} \right)} \right]}^\prime }}}}$
$\begin{array}{*{20}{l}} {}\\ {{{\left[ {n\left( {n + 2} \right){x^{n + 1}} - {{\left( {n + 1} \right)}^2}{x^n} + 1} \right]}^\prime } = n\left( {n + 2} \right)\left( {n + 1} \right){x^n} - n{{\left( {n + 1} \right)}^2}{x^{n - 1}} = n\left( {n + 1} \right){x^{n - 1}}\left[ {\left( {n + 2} \right)x - \left( {n + 1} \right)} \right]}\\ {{{\left[ {\left( {x - 1} \right)} \right]}^\prime } = 1}\\ {L = \frac{1}{2} \times \mathop {{\rm{lim}}}\limits_{x \to 1} \frac{{n\left( {n + 1} \right){x^{n - 1}}\left[ {\left( {n + 2} \right)x - \left( {n + 1} \right)} \right]}}{1} = \frac{1}{2}n\left( {n + 1} \right)} \end{array}$

Regulile lui l`Hospital

Regulile lui l`Hospital

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