Límite de la función n*log(n)*log(log(n))

Tenemos la indeterminación de tipo

0/0,

tal que el límite para el numerador es
$$\lim_{n \to 1^+}\left(n \log{\left(n \right)}\right) = 0$$
y el límite para el denominador es
$$\lim_{n \to 1^+} \frac{1}{\log{\left(\log{\left(n \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to 1^+}\left(n \log{\left(n \right)} \log{\left(\log{\left(n \right)} \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{n \to 1^+}\left(n \log{\left(n \right)} \log{\left(\log{\left(n \right)} \right)}\right)$$
=
$$\lim_{n \to 1^+}\left(\frac{\frac{d}{d n} n \log{\left(n \right)}}{\frac{d}{d n} \frac{1}{\log{\left(\log{\left(n \right)} \right)}}}\right)$$
=
$$\lim_{n \to 1^+}\left(- n \left(\log{\left(n \right)} + 1\right) \log{\left(n \right)} \log{\left(\log{\left(n \right)} \right)}^{2}\right)$$
=
$$\lim_{n \to 1^+}\left(- \left(\log{\left(n \right)} + 1\right) \log{\left(n \right)} \log{\left(\log{\left(n \right)} \right)}^{2}\right)$$
=
$$\lim_{n \to 1^+}\left(\frac{\frac{d}{d n} \left(- \left(\log{\left(n \right)} + 1\right) \log{\left(n \right)}\right)}{\frac{d}{d n} \frac{1}{\log{\left(\log{\left(n \right)} \right)}^{2}}}\right)$$
=
$$\lim_{n \to 1^+}\left(- \frac{n \left(\frac{- \log{\left(n \right)} - 1}{n} - \frac{\log{\left(n \right)}}{n}\right) \log{\left(n \right)} \log{\left(\log{\left(n \right)} \right)}^{3}}{2}\right)$$
=
$$\lim_{n \to 1^+}\left(- \frac{\left(\frac{- \log{\left(n \right)} - 1}{n} - \frac{\log{\left(n \right)}}{n}\right) \log{\left(n \right)} \log{\left(\log{\left(n \right)} \right)}^{3}}{2}\right)$$
=
$$\lim_{n \to 1^+}\left(\frac{\frac{d}{d n} \left(- \frac{\log{\left(n \right)} \log{\left(\log{\left(n \right)} \right)}^{3}}{2}\right)}{\frac{d}{d n} \frac{1}{\frac{- \log{\left(n \right)} - 1}{n} - \frac{\log{\left(n \right)}}{n}}}\right)$$
=
$$\lim_{n \to 1^+}\left(\frac{\left(\frac{- \log{\left(n \right)} - 1}{n} - \frac{\log{\left(n \right)}}{n}\right)^{2} \left(- \frac{\log{\left(\log{\left(n \right)} \right)}^{3}}{2 n} - \frac{3 \log{\left(\log{\left(n \right)} \right)}^{2}}{2 n}\right)}{\frac{- \log{\left(n \right)} - 1}{n^{2}} - \frac{\log{\left(n \right)}}{n^{2}} + \frac{2}{n^{2}}}\right)$$
=
$$\lim_{n \to 1^+}\left(- \frac{\log{\left(\log{\left(n \right)} \right)}^{3}}{2 n} - \frac{3 \log{\left(\log{\left(n \right)} \right)}^{2}}{2 n}\right)$$
=
$$\lim_{n \to 1^+}\left(- \frac{\log{\left(\log{\left(n \right)} \right)}^{3}}{2 n} - \frac{3 \log{\left(\log{\left(n \right)} \right)}^{2}}{2 n}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)