Question

determine the following limit.

\\\lim_{x \to -\infty} \frac{\sqrt{144x^2 + x}}{x}\\

select the correct choice below, and, if necessary, fill in the answer box to complete your choice.

a. \\\lim_{x \to -\infty} \frac{\sqrt{144x^2 + x}}{x} = \quad\\ (simplify your answer.)
b. the limit does not exist and is neither \\(\infty\\) nor \\(-\infty\\).

Explanation:

Divide numerator and denominator by x

$$ \lim_{x \to -\infty} \frac{\sqrt{144x^2 + x}}{x} $$

Since \(x < 0\), we have \(x = -\sqrt{x^2}\).

$$ \lim_{x \to -\infty} \frac{\sqrt{144x^2 + x}}{-\sqrt{x^2}} = \lim_{x \to -\infty} -\sqrt{\frac{144x^2 + x}{x^2}} $$

Simplify the expression inside the radical

$$ -\sqrt{\frac{144x^2 + x}{x^2}} = -\sqrt{144 + \frac{1}{x}} $$

Evaluate the limit

$$ \lim_{x \to -\infty} -\sqrt{144 + \frac{1}{x}} = -\sqrt{144 + 0} = -12 $$

Answer:

• A. \(\lim_{x \to -\infty} \frac{\sqrt{144x^2 + x}}{x} = -12\) (Correct answer)
• B. The limit does not exist and is neither \(\infty\) nor \(-\infty\).