Questions: Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim as x approaches -∞ of (3x) / sqrt(x^2 - x)

Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim as x approaches -∞ of (3x) / sqrt(x^2 - x)
Transcript text: Find the limit, if it exists. (If an answer does not exist, enter DNE.) \[ \lim _{x \rightarrow-\infty} \frac{3 x}{\sqrt{x^{2}-x}} \] $\square$
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Solution

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Solution Steps

To find the limit of the given expression as \( x \) approaches \(-\infty\), we can simplify the expression by dividing the numerator and the denominator by \( x \). This will help us analyze the behavior of the function as \( x \) becomes very large in the negative direction. After simplification, we can evaluate the limit of the resulting expression.

Step 1: Simplifying the Expression

We start with the limit: \[ \lim_{x \rightarrow -\infty} \frac{3x}{\sqrt{x^2 - x}} \] To simplify, we divide the numerator and the denominator by \( x \): \[ \frac{3x}{\sqrt{x^2 - x}} = \frac{3}{\sqrt{1 - \frac{1}{x}}} \]

Step 2: Evaluating the Limit

As \( x \) approaches \(-\infty\), the term \(\frac{1}{x}\) approaches \(0\). Thus, the expression simplifies to: \[ \lim_{x \rightarrow -\infty} \frac{3}{\sqrt{1 - 0}} = \frac{3}{\sqrt{1}} = 3 \] However, since \( x \) is negative, we need to consider the sign of the square root: \[ \sqrt{x^2 - x} \approx |x| \text{ for large } |x| \text{, which is } -x \text{ when } x \text{ is negative.} \] Thus, the limit becomes: \[ \lim_{x \rightarrow -\infty} \frac{3x}{-x} = -3 \]

Final Answer

The limit exists and is given by: \[ \boxed{-3} \]

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