Scriem [tex]x^4\left(e^{\frac{1}{x^2+1}}-e^{\frac{1}{x^2}}\right)=e^{\frac{1}{x^2}}\cdot\frac{e^{\frac{1}{x^2+1}-\frac{1}{x^2}}-1}{\frac{1}{x^4}}=e^{\frac{1}{x^2}}\cdot\frac{e^{\frac{-1}{x^4+x^2}}-1}{\frac{1}{x^4}}[/tex]
Apoi, [tex]e^\frac{1}{x^2}\rightarrow e^0=1,[/tex] iar [tex]\frac{e^{\frac{-1}{x^4+x^2}}-1}{\frac{1}{x^4}}=\frac{e^{\frac{-1}{x^4+x^2}}-1}{\frac{-1}{x^4+x^2}}\cdot\frac{-x^4}{x^4+x^2}\rightarrow1\cdot (-1)=-1[/tex].
Am folosit limita fundamentala [tex]\lim_{x\to 0}\frac{e^x-1}{x}=1.[/tex]
