Evaluate the limit. $$ \lim _{x \rightarrow-\infty} \frac{x^{3}-x^{2}-9 x+2}{1-6 x^{2}} $$

Question

UpStudy AI · Accepted Answer

Calculate the limit $$ \lim_{x\rightarrow -\infty} \frac{x^{3}-x^{2}-9x+2}{1-6x^{2}} $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{x\rightarrow -\infty}\left(\frac{x^{3}-x^{2}-9x+2}{1-6x^{2}}\right)$$
- step1: Rewrite the expression:
  $$\lim _{x\rightarrow -\infty}\left(\frac{x^{3}\left(1-\frac{1}{x}-\frac{9}{x^{2}}+\frac{2}{x^{3}}\right)}{x^{2}\left(\frac{1}{x^{2}}-6\right)}\right)$$
- step2: Reduce the fraction:
  $$\lim _{x\rightarrow -\infty}\left(\frac{x\left(1-\frac{1}{x}-\frac{9}{x^{2}}+\frac{2}{x^{3}}\right)}{\frac{1}{x^{2}}-6}\right)$$
- step3: Rewrite the expression:
  $$\frac{\lim _{x\rightarrow -\infty}\left(x\left(1-\frac{1}{x}-\frac{9}{x^{2}}+\frac{2}{x^{3}}\right)\right)}{\lim _{x\rightarrow -\infty}\left(\frac{1}{x^{2}}-6\right)}$$
- step4: Calculate:
  $$\frac{-\infty}{\lim _{x\rightarrow -\infty}\left(\frac{1}{x^{2}}-6\right)}$$
- step5: Calculate:
  $$\frac{-\infty}{-6}$$
- step6: Calculate:
  $$+\infty$$
The limit of the expression $$\frac{x^{3}-x^{2}-9x+2}{1-6x^{2}}$$ as $$x$$ approaches negative infinity is $$+\infty$$.
The limit is $$+\infty$$.

Evaluate the limit.

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Calculus Questions