Find the limit. Use a graphing utility to verify your result. (Hint: Treat the expression as a fraction whose denominator is 1 , and rationalize the
numerator. If an answer does not exist, enter DNE.)

Ask by Burgess Maxwell. in the United States

Mar 31,2025

Question

Find the limit. Use a graphing utility to verify your result. (Hint: Treat the expression as a fraction whose denominator is 1 , and rationalize the
numerator. If an answer does not exist, enter DNE.)

Ask by Burgess Maxwell. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

Calculate the limit $$ \lim_{x ightarrow -\infty} (9x+\sqrt{81x^2-x}) $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{xightarrow -\infty}\left(9x+\sqrt{81x^{2}-x}ight)$$
- step1: Multiply by the Conjugate:
  $$\lim _{xightarrow -\infty}\left(\frac{x}{9x-\sqrt{81x^{2}-x}}ight)$$
- step2: Rearrange the terms:
  $$\lim _{xightarrow -\infty}\left(\frac{1	imes x}{\left(9+\sqrt{81-\frac{1}{x}}ight)x}ight)$$
- step3: Reduce the fraction:
  $$\lim _{xightarrow -\infty}\left(\frac{1}{9+\sqrt{81-\frac{1}{x}}}ight)$$
- step4: Rewrite the expression:
  $$\frac{\lim _{xightarrow -\infty}\left(1ight)}{\lim _{xightarrow -\infty}\left(9+\sqrt{81-\frac{1}{x}}ight)}$$
- step5: Evaluate:
  $$\frac{1}{\lim _{xightarrow -\infty}\left(9+\sqrt{81-\frac{1}{x}}ight)}$$
- step6: Evaluate:
  $$\frac{1}{18}$$
The limit of the expression $$9x+\sqrt{81x^{2}-x}$$ as $$x$$ approaches negative infinity is $$\frac{1}{18}$$ or $$0.0\dot{5}$$.

To verify this result, you can use a graphing utility to plot the function and observe the behavior of the function as $$x$$ approaches negative infinity.
The limit is $$ \frac{1}{18} $$ or approximately $$ 0.0555 $$.

Find the limit. Use a graphing utility to verify your result. (Hint: Treat the expression as a fraction whose denominator is 1 , and rationalize the
numerator. If an answer does not exist, enter DNE.)

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