Răspuns:
[tex]\displaystyle\lim_{x\nearrow 0}\displaystyle\frac{3^{\frac{1}{x}}}{x}=\lim_{x\nearrow 0}\frac{1}{x}\cdot3^{\frac{1}{x}[/tex]
Notăm [tex]\frac{1}{x}=y\Rightarrow y\to-\infty[/tex]
Atunci limita devine
[tex]\displaystyle\lim_{y\to-\infty}y\cdot 3^y=\lim_{y\to-\infty}\frac{y}{3^{-y}}=\lim_{y\to-\infty}\frac{1}{-3^{-y}\ln 3}=0[/tex]
(Am folosit l'Hospital)
[tex]\displaystyle\lim_{x\searrow 0}\frac{3^{\frac{1}{x}}}{x}=\lim_{x\searrow 0}\frac{1}{x}\cdot 3^{\frac{1}{x}}=\infty\cdot\infty=\infty[/tex]
Explicație pas cu pas:
