Explicație pas cu pas:
[tex] lim_{x - > 0 + }( \frac{ {x}^{2} }{{e}^{ - \frac{1}{x} } }) = \\[/tex]
[tex]= lim_{x - > 0 + }( {e}^{ \frac{1}{x}} {x}^{2} ) \\[/tex]
[tex]= lim_{x - > 0 + }( \frac{ {e}^{ \frac{1}{x} } }{ \frac{1}{ {x}^{2} } } ) \\ [/tex]
=> l'Hospital
[tex] = lim_{x - > 0 + }( \frac{-\frac{ {e}^{ \frac{1}{x} } }{ {x}^{2} } }{ - \frac{2}{ {x}^{3} } } ) \\[/tex]
[tex]= lim_{x - > 0 + }( \frac{x {e}^{ \frac{1}{x} } }{2} ) \\ [/tex]
[tex]= \frac{1}{2}lim_{x - > 0 + }(x {e}^{ \frac{1}{x} } ) \\ [/tex]
[tex]= \frac{1}{2}lim_{x - > 0 + }( \frac{ {e}^{ \frac{1}{x} } }{ \frac{1}{x} } )\\ [/tex]
=> l'Hospital
[tex]= \frac{1}{2} lim_{x - > 0 + }( \frac{ - \frac{ {e}^{ \frac{1}{x} } }{ {x}^{2} } }{ - \frac{1}{ {x}^{2} } } ) \\ [/tex]
[tex]= \frac{1}{2}lim_{x - > 0 + }( {e}^{ \frac{1}{x} } ) \\[/tex]
[tex]= \frac{1}{2} \times \infty = \infty \\ [/tex]
