$\lim _{x \rightarrow 2} \frac{x+3 x^2+5 x^3+7 x^4-166}{x-2}=$

The limit is of the indeterminate form $\frac{0}{0}$ when $x = 2$, so L'Hôpital's Rule applies.

Applying L'Hôpital's Rule: Differentiate the numerator and denominator separately.

The derivative of the numerator $x + 3x^2 + 5x^3 + 7x^4 - 166$ is $1 + 6x + 15x^2 + 28x^3$.

The derivative of the denominator $x - 2$ is $1$.

The limit becomes $\lim_{x \to 2} (1 + 6x + 15x^2 + 28x^3)$.

Evaluating at $x = 2$: $1 + 12 + 60 + 224 = 297$.

The limit is $\boxed{297}$.