\( \lim _{x \rightarrow-=} \frac{x^{2}-x^{2}+x}{x^{3}-x+5} \)

Calculate the limit \( \lim_{x\rightarrow -\infty} \frac{x^{2}-x^{2}+x}{x^{3}-x+5} \). Evaluate the limit by following steps: - step0: Evaluate using transformations: \(\lim _{x\rightarrow -\infty}\left(\frac{x^{2}-x^{2}+x}{x^{3}-x+5}\right)\) - step1: Calculate: \(\lim _{x\rightarrow -\infty}\left(\frac{x}{x^{3}-x+5}\right)\) - step2: Rewrite the expression: \(\lim _{x\rightarrow -\infty}\left(\frac{x}{x^{3}}\times \frac{1}{1-\frac{1}{x^{2}}+\frac{5}{x^{3}}}\right)\) - step3: Calculate: \(\lim _{x\rightarrow -\infty}\left(\frac{1}{x^{2}}\times \frac{1}{1-\frac{1}{x^{2}}+\frac{5}{x^{3}}}\right)\) - step4: Rewrite the expression: \(\lim _{x\rightarrow -\infty}\left(\frac{1}{x^{2}}\right)\times \lim _{x\rightarrow -\infty}\left(\frac{1}{1-\frac{1}{x^{2}}+\frac{5}{x^{3}}}\right)\) - step5: Calculate: \(\lim _{x\rightarrow -\infty}\left(\frac{1}{x^{2}}\right)\times \frac{\lim _{x\rightarrow -\infty}\left(1\right)}{\lim _{x\rightarrow -\infty}\left(1-\frac{1}{x^{2}}+\frac{5}{x^{3}}\right)}\) - step6: Rewrite the expression: \(\lim _{x\rightarrow -\infty}\left(\frac{1}{x^{2}}\right)\times \frac{1}{\lim _{x\rightarrow -\infty}\left(1-\frac{1}{x^{2}}+\frac{5}{x^{3}}\right)}\) - step7: Rewrite the expression: \(\lim _{x\rightarrow -\infty}\left(\frac{1}{x^{2}}\right)\times \frac{1}{1}\) - step8: Rewrite the expression: \(0\times \frac{1}{1}\) - step9: Calculate: \(0\times 1\) - step10: Calculate: \(0\) The limit of the expression \( \frac{x^{2}-x^{2}+x}{x^{3}-x+5} \) as \( x \) approaches negative infinity is 0.