Límite de la función x*(-log(2*x)+log(1+2*x))

Tenemos la indeterminación de tipo

oo/oo,

tal que el límite para el numerador es
$$\lim_{x \to \infty} x = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} \frac{1}{- \log{\left(2 x \right)} + \log{\left(2 x + 1 \right)}} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(x \left(- \log{\left(2 x \right)} + \log{\left(2 x + 1 \right)}\right)\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \frac{1}{- \log{\left(2 x \right)} + \log{\left(2 x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(- \log{\left(2 x \right)} + \log{\left(2 x + 1 \right)}\right)^{2}}{- \frac{2}{2 x + 1} + \frac{1}{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{2}{2 x + 1} + \frac{1}{x}}}{\frac{d}{d x} \frac{1}{\left(- \log{\left(2 x \right)} + \log{\left(2 x + 1 \right)}\right)^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{\left(- \frac{4}{2 x + 1} + \frac{2}{x}\right) \left(\frac{4}{\frac{16 x^{2} \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{48 x^{2} \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{48 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{48 x^{2} \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{96 x^{2} \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{48 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{16 x^{2} \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{48 x^{2} \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{48 x^{2} \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{16 x^{2} \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{16 x \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{48 x \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{48 x \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{48 x \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{96 x \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{48 x \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{16 x \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{48 x \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{48 x \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{16 x \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} - 4 \log{\left(x \right)}^{3} + 12 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)} - 12 \log{\left(2 \right)} \log{\left(x \right)}^{2} - 12 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2} + 24 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)} - 12 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 4 \log{\left(2 x + 1 \right)}^{3} - 12 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2} + 12 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)} - 4 \log{\left(2 \right)}^{3} + \frac{4 \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{12 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{12 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{24 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{4 \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{12 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{12 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{4 \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{4 \log{\left(x \right)}^{3}}{x} + \frac{12 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{x} - \frac{12 \log{\left(2 \right)} \log{\left(x \right)}^{2}}{x} - \frac{12 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{x} + \frac{24 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{x} - \frac{12 \log{\left(2 \right)}^{2} \log{\left(x \right)}}{x} + \frac{4 \log{\left(2 x + 1 \right)}^{3}}{x} - \frac{12 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{x} + \frac{12 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{x} - \frac{4 \log{\left(2 \right)}^{3}}{x} - \frac{\log{\left(x \right)}^{3}}{x^{2}} + \frac{3 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{x^{2}} - \frac{3 \log{\left(2 \right)} \log{\left(x \right)}^{2}}{x^{2}} - \frac{3 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{x^{2}} + \frac{6 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{x^{2}} - \frac{3 \log{\left(2 \right)}^{2} \log{\left(x \right)}}{x^{2}} + \frac{\log{\left(2 x + 1 \right)}^{3}}{x^{2}} - \frac{3 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{x^{2}} + \frac{3 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{x^{2}} - \frac{\log{\left(2 \right)}^{3}}{x^{2}}} - \frac{4}{\frac{8 x^{2} \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{24 x^{2} \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{24 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{24 x^{2} \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{48 x^{2} \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{24 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{8 x^{2} \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{24 x^{2} \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{24 x^{2} \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{8 x^{2} \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{4 x \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{12 x \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 x \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{12 x \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{24 x \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 x \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{4 x \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{12 x \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{12 x \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{4 x \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} - 2 \log{\left(x \right)}^{3} + 6 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)} - 6 \log{\left(2 \right)} \log{\left(x \right)}^{2} - 6 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2} + 12 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 x + 1 \right)}^{3} - 6 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2} + 6 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)} - 2 \log{\left(2 \right)}^{3} - \frac{\log{\left(x \right)}^{3}}{x} + \frac{3 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{x} - \frac{3 \log{\left(2 \right)} \log{\left(x \right)}^{2}}{x} - \frac{3 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{x} + \frac{6 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{x} - \frac{3 \log{\left(2 \right)}^{2} \log{\left(x \right)}}{x} + \frac{\log{\left(2 x + 1 \right)}^{3}}{x} - \frac{3 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{x} + \frac{3 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{x} - \frac{\log{\left(2 \right)}^{3}}{x}} + \frac{1}{\frac{4 x^{2} \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{12 x^{2} \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{12 x^{2} \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{24 x^{2} \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{4 x^{2} \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{12 x^{2} \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{12 x^{2} \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{4 x^{2} \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} - \log{\left(x \right)}^{3} + 3 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)} - 3 \log{\left(2 \right)} \log{\left(x \right)}^{2} - 3 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2} + 6 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)} - 3 \log{\left(2 \right)}^{2} \log{\left(x \right)} + \log{\left(2 x + 1 \right)}^{3} - 3 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2} + 3 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)} - \log{\left(2 \right)}^{3}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{\left(- \frac{4}{2 x + 1} + \frac{2}{x}\right) \left(\frac{4}{\frac{16 x^{2} \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{48 x^{2} \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{48 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{48 x^{2} \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{96 x^{2} \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{48 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{16 x^{2} \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{48 x^{2} \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{48 x^{2} \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{16 x^{2} \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{16 x \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{48 x \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{48 x \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{48 x \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{96 x \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{48 x \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{16 x \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{48 x \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{48 x \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{16 x \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} - 4 \log{\left(x \right)}^{3} + 12 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)} - 12 \log{\left(2 \right)} \log{\left(x \right)}^{2} - 12 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2} + 24 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)} - 12 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 4 \log{\left(2 x + 1 \right)}^{3} - 12 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2} + 12 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)} - 4 \log{\left(2 \right)}^{3} + \frac{4 \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{12 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{12 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{24 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{4 \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{12 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{12 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{4 \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{4 \log{\left(x \right)}^{3}}{x} + \frac{12 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{x} - \frac{12 \log{\left(2 \right)} \log{\left(x \right)}^{2}}{x} - \frac{12 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{x} + \frac{24 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{x} - \frac{12 \log{\left(2 \right)}^{2} \log{\left(x \right)}}{x} + \frac{4 \log{\left(2 x + 1 \right)}^{3}}{x} - \frac{12 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{x} + \frac{12 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{x} - \frac{4 \log{\left(2 \right)}^{3}}{x} - \frac{\log{\left(x \right)}^{3}}{x^{2}} + \frac{3 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{x^{2}} - \frac{3 \log{\left(2 \right)} \log{\left(x \right)}^{2}}{x^{2}} - \frac{3 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{x^{2}} + \frac{6 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{x^{2}} - \frac{3 \log{\left(2 \right)}^{2} \log{\left(x \right)}}{x^{2}} + \frac{\log{\left(2 x + 1 \right)}^{3}}{x^{2}} - \frac{3 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{x^{2}} + \frac{3 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{x^{2}} - \frac{\log{\left(2 \right)}^{3}}{x^{2}}} - \frac{4}{\frac{8 x^{2} \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{24 x^{2} \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{24 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{24 x^{2} \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{48 x^{2} \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{24 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{8 x^{2} \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{24 x^{2} \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{24 x^{2} \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{8 x^{2} \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{4 x \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{12 x \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 x \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{12 x \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{24 x \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 x \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{4 x \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{12 x \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{12 x \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{4 x \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} - 2 \log{\left(x \right)}^{3} + 6 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)} - 6 \log{\left(2 \right)} \log{\left(x \right)}^{2} - 6 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2} + 12 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)} - 6 \log{\left(2 \right)}^{2} \log{\left(x \right)} + 2 \log{\left(2 x + 1 \right)}^{3} - 6 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2} + 6 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)} - 2 \log{\left(2 \right)}^{3} - \frac{\log{\left(x \right)}^{3}}{x} + \frac{3 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{x} - \frac{3 \log{\left(2 \right)} \log{\left(x \right)}^{2}}{x} - \frac{3 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{x} + \frac{6 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{x} - \frac{3 \log{\left(2 \right)}^{2} \log{\left(x \right)}}{x} + \frac{\log{\left(2 x + 1 \right)}^{3}}{x} - \frac{3 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{x} + \frac{3 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{x} - \frac{\log{\left(2 \right)}^{3}}{x}} + \frac{1}{\frac{4 x^{2} \log{\left(x \right)}^{3}}{4 x^{2} + 4 x + 1} - \frac{12 x^{2} \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2} + 4 x + 1} + \frac{12 x^{2} \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{24 x^{2} \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{12 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{2} + 4 x + 1} - \frac{4 x^{2} \log{\left(2 x + 1 \right)}^{3}}{4 x^{2} + 4 x + 1} + \frac{12 x^{2} \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2}}{4 x^{2} + 4 x + 1} - \frac{12 x^{2} \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)}}{4 x^{2} + 4 x + 1} + \frac{4 x^{2} \log{\left(2 \right)}^{3}}{4 x^{2} + 4 x + 1} - \log{\left(x \right)}^{3} + 3 \log{\left(x \right)}^{2} \log{\left(2 x + 1 \right)} - 3 \log{\left(2 \right)} \log{\left(x \right)}^{2} - 3 \log{\left(x \right)} \log{\left(2 x + 1 \right)}^{2} + 6 \log{\left(2 \right)} \log{\left(x \right)} \log{\left(2 x + 1 \right)} - 3 \log{\left(2 \right)}^{2} \log{\left(x \right)} + \log{\left(2 x + 1 \right)}^{3} - 3 \log{\left(2 \right)} \log{\left(2 x + 1 \right)}^{2} + 3 \log{\left(2 \right)}^{2} \log{\left(2 x + 1 \right)} - \log{\left(2 \right)}^{3}}\right)}\right)$$
=
$$\frac{1}{2}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)