Question

Limita asta vă rog……… nu îmi iese cu L’Hospital

Answer 1

Explicație pas cu pas:

$lim_{x - > 0 + }( \frac{ {x}^{2} }{{e}^{ - \frac{1}{x} } }) =$

$= lim_{x - > 0 + }( {e}^{ \frac{1}{x}} {x}^{2} )$

$= lim_{x - > 0 + }( \frac{ {e}^{ \frac{1}{x} } }{ \frac{1}{ {x}^{2} } } )$

=> l'Hospital

$= lim_{x - > 0 + }( \frac{-\frac{ {e}^{ \frac{1}{x} } }{ {x}^{2} } }{ - \frac{2}{ {x}^{3} } } )$

$= lim_{x - > 0 + }( \frac{x {e}^{ \frac{1}{x} } }{2} )$

$= \frac{1}{2}lim_{x - > 0 + }(x {e}^{ \frac{1}{x} } )$

$= \frac{1}{2}lim_{x - > 0 + }( \frac{ {e}^{ \frac{1}{x} } }{ \frac{1}{x} } )$

=> l'Hospital

$= \frac{1}{2} lim_{x - > 0 + }( \frac{ - \frac{ {e}^{ \frac{1}{x} } }{ {x}^{2} } }{ - \frac{1}{ {x}^{2} } } )$

$= \frac{1}{2}lim_{x - > 0 + }( {e}^{ \frac{1}{x} } )$

$= \frac{1}{2} \times \infty = \infty$