2. \( \lim _{x \rightarrow-\infty} \frac{-12 x^{11}+2 x+3}{x^{6}+x^{2}-10 x+81} \)

To find the limit \( \lim _{x \rightarrow-\infty} \frac{-12 x^{11}+2 x+3}{x^{6}+x^{2}-10 x+81} \), we can analyze the highest degree terms in the numerator and denominator. In the numerator, the highest degree term is \(-12 x^{11}\). In the denominator, the highest degree term is \(x^6\). As \(x\) approaches \(-\infty\), the behavior of the function will be dominated by these highest degree terms. Thus, we can simplify the limit as follows: \[ \lim_{x \to -\infty} \frac{-12 x^{11}}{x^6} = \lim_{x \to -\infty} -12 x^{5} \] Now, as \(x\) approaches \(-\infty\), \(x^5\) will also approach \(-\infty\). Therefore, we find: \[ \lim_{x \to -\infty} -12 x^{5} = \infty \] Thus, the final answer is: \[ \boxed{\infty} \]