Tomamos como el límite
$$\lim_{x \to 0^+}\left(\frac{9 x + \left(x^{3} + 7\right)}{9 x^{2} + \left(6 x + 1\right)}\right)$$
cambiamos
$$\lim_{x \to 0^+}\left(\frac{9 x + \left(x^{3} + 7\right)}{9 x^{2} + \left(6 x + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x^{3} + 9 x + 7}{\left(3 x + 1\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x^{3} + 9 x + 7}{\left(3 x + 1\right)^{2}}\right) = $$
$$\frac{0^{3} + 0 \cdot 9 + 7}{\left(0 \cdot 3 + 1\right)^{2}} = $$
= 7
Entonces la respuesta definitiva es:
$$\lim_{x \to 0^+}\left(\frac{9 x + \left(x^{3} + 7\right)}{9 x^{2} + \left(6 x + 1\right)}\right) = 7$$