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

Evaluate the limit by following steps: - step0: Evaluate using transformations: \(\lim _{x\rightarrow -\infty}\left(\frac{-12x^{11}+2x+3}{x^{6}+x^{2}-10x+81}\right)\) - step1: Rewrite the expression: \(\lim _{x\rightarrow -\infty}\left(\frac{x^{11}\left(-12+\frac{2}{x^{10}}+\frac{3}{x^{11}}\right)}{x^{6}\left(1+\frac{1}{x^{4}}-\frac{10}{x^{5}}+\frac{81}{x^{6}}\right)}\right)\) - step2: Reduce the fraction: \(\lim _{x\rightarrow -\infty}\left(\frac{x^{5}\left(-12+\frac{2}{x^{10}}+\frac{3}{x^{11}}\right)}{1+\frac{1}{x^{4}}-\frac{10}{x^{5}}+\frac{81}{x^{6}}}\right)\) - step3: Rewrite the expression: \(\frac{\lim _{x\rightarrow -\infty}\left(x^{5}\left(-12+\frac{2}{x^{10}}+\frac{3}{x^{11}}\right)\right)}{\lim _{x\rightarrow -\infty}\left(1+\frac{1}{x^{4}}-\frac{10}{x^{5}}+\frac{81}{x^{6}}\right)}\) - step4: Calculate: \(\frac{+\infty}{\lim _{x\rightarrow -\infty}\left(1+\frac{1}{x^{4}}-\frac{10}{x^{5}}+\frac{81}{x^{6}}\right)}\) - step5: Calculate: \(\frac{+\infty}{1}\) - step6: Calculate: \(+\infty\) La limite de l'expression donnée est : \[ \lim_{x \rightarrow -\infty} \frac{-12 x^{11}+2 x+3}{x^{6}+x^{2}-10 x+81} = +\infty \] Cela signifie que lorsque \( x \) tend vers moins l'infini, la valeur de l'expression augmente sans limite.