The value of K

[EstudanteQuimica]

Someone can help with this question ? (the value of K)

 

[Dr.Peterson]

I would apply l'Hopital to the log of the limit.

 

[EstudanteQuimica]

How?

 

[skeeter]

[math]\left(1+\dfrac{k+20}{x} + \dfrac{20k}{x^2}\right)^x[/math]
[MATH]\left(\dfrac{(x+k)(x+20)}{x^2}\right)^x[/MATH]
[MATH]\left(\dfrac{x+k}{x} \cdot \dfrac{x+20}{x}\right)^x[/MATH]
[MATH]\left(\dfrac{x+k}{x}\right)^x \cdot \left(\dfrac{x+20}{x}\right)^x[/MATH]
[MATH]\lim_{x \to \infty}\left(\dfrac{x+k}{x}\right)^x \cdot \lim_{x \to \infty} \left(\dfrac{x+20}{x}\right)^x[/MATH]
[MATH]e^k \cdot e^{20} = e^{60}[/MATH]