[m]\frac{1}{x}=t[/m] ⇒ [m]x → ∞[/m] ⇒ [m]t → 0[/m]
[m]x=\frac{1}{t}[/m]
[m]x+1=1+\frac{1}{t}[/m] ⇒[m]\frac{1}{x+1}=\frac{t}{t+1}[/m]
[m]lim_{t → 0 }\frac{a^{\frac{t}{t+1}}\cdot (a^{\frac{t^2}{(t+1)}}-1)}{t^2}=lim_{t → 0 }a^{\frac{t}{t+1}}\cdot \frac{a^{\frac{t^2}{(t+1)}}-1}{t^2}=lim_{t → 0 }a^{\frac{t}{t+1}}\cdot lim_{t → 0 }\frac{a^{\frac{t^2}{(t+1)}}-1}{t^2}=1\cdot lim_{t → 0 }\frac{a^{\frac{t^2}{(t+1)}}-1}{t^2}= lim_{t → 0 }\frac{\frac{t^2}{t+1}\cdot lna}{t^2}=lna[/m]