Tenemos la indeterminación de tipo
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \tan{\left(3 x^{2} - 5 x \right)} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(- \frac{1}{\cot{\left(3 x \right)}}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\tan{\left(3 x^{2} - 5 x \right)} \left(- \cot{\left(3 x \right)}\right)\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(- \tan{\left(x \left(3 x - 5\right) \right)} \cot{\left(3 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(3 x^{2} - 5 x \right)}}{\frac{d}{d x} \left(- \frac{1}{\cot{\left(3 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(6 x - 5\right) \left(\tan^{2}{\left(3 x^{2} - 5 x \right)} + 1\right) \cot^{2}{\left(3 x \right)}}{3 \cot^{2}{\left(3 x \right)} + 3}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cot^{2}{\left(3 x \right)}}{3 \cot^{2}{\left(3 x \right)} + 3}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{3 \cot^{2}{\left(3 x \right)} + 3}}{\frac{d}{d x} \frac{1}{5 \cot^{2}{\left(3 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(6 \cot^{2}{\left(3 x \right)} + 6\right) \left(- \frac{9 \cot^{4}{\left(3 x \right)}}{- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}} - \frac{18 \cot^{2}{\left(3 x \right)}}{- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}} - \frac{9}{- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{6 \cot^{2}{\left(3 x \right)} + 6}}{\frac{d}{d x} \left(- \frac{9 \cot^{4}{\left(3 x \right)}}{- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}} - \frac{18 \cot^{2}{\left(3 x \right)}}{- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}} - \frac{9}{- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{6 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(6 \cot^{2}{\left(3 x \right)} + 6\right)^{2} \left(- \frac{9 \left(90 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 90 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{4}{\left(3 x \right)}}{\left(- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{18 \left(90 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 90 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{2}{\left(3 x \right)}}{\left(- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(90 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 90 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right)}{\left(- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}}{- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}} - \frac{18 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{6 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{\left(6 \cot^{2}{\left(3 x \right)} + 6\right)^{2} \left(- \frac{9 \left(90 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 90 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{4}{\left(3 x \right)}}{\left(- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{18 \left(90 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 90 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right) \cot^{2}{\left(3 x \right)}}{\left(- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(90 \left(- 18 \cot^{2}{\left(3 x \right)} - 18\right) \cot^{5}{\left(3 x \right)} + 90 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}\right)}{\left(- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}\right)^{2}} - \frac{9 \left(- 12 \cot^{2}{\left(3 x \right)} - 12\right) \cot^{3}{\left(3 x \right)}}{- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}} - \frac{18 \left(- 6 \cot^{2}{\left(3 x \right)} - 6\right) \cot{\left(3 x \right)}}{- 90 \cot^{6}{\left(3 x \right)} - 90 \cot^{4}{\left(3 x \right)}}\right)}\right)$$
=
$$\frac{5}{3}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)