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Let's solve the given limit problem:

The expression to evaluate is:

\[
\lim_{x \to -\infty} \left( 5x + \sqrt{25x^2 - x} \right)
\]

### Step 1: Simplify the square root term

Inside the square root, factor \( x^2 \) from the expression:

\[
\sqrt{25x^2 - x} = \sqrt{x^2(25 - \frac{1}{x})} = |x|\sqrt{25 - \frac{1}{x}}
\]

Since \( x \to -\infty \), we know \( |x| = -x \), so:

\[
\sqrt{25x^2 - x} = -x\sqrt{25 - \frac{1}{x}}
\]

### Step 2: Rewrite the limit expression

Now substitute this back into the limit expression:

\[
5x + \sqrt{25x^2 - x} = 5x - x\sqrt{25 - \frac{1}{x}}
\]

### Step 3: Factor out \( x \)

Factor \( x \) from the expression:

\[
x \left( 5 - \sqrt{25 - \frac{1}{x}} \right)
\]

### Step 4: Evaluate the limit as \( x \to -\infty \)

As \( x \to -\infty \), the term \( \frac{1}{x} \to 0 \), so the expression becomes:

\[
\lim_{x \to -\infty} x \left( 5 - \sqrt{25} \right) = \lim_{x \to -\infty} x(5 - 5) = \lim_{x \to -\infty} x(0) = 0
\]

Thus, the limit is:

\[
\boxed{0}
\]

### Explanation for incorrect entry

You entered \( 0 \), which seems correct based on the analysis above. However, if your answer was marked incorrect, it's possible there might have been a typo or input issue. The mathematical solution is indeed correct.

Would you like a detailed explanation on any step? Or perhaps more practice problems on limits?

### Further Questions:
1. What happens if \( x \to +\infty \) for this limit problem?
2. Can this method be generalized for other similar radical expressions?
3. How would you rationalize the numerator if there was a negative sign in front of the square root?
4. How can graphing utilities help in verifying such limits?
5. How can this approach be extended to higher powers of \( x \)?

**Tip:** When evaluating limits at infinity, always check the dominant terms in both the numerator and denominator (if applicable).

Find the limit: lim x -> -∞ (5x + sqrt(25x^2 - x))

Learn how to find the limit of the expression 5x + sqrt(25x^2 - x) as x approaches negative infinity using factoring and limit laws. This problem involves simplifying square roots and evaluating limits at infinity, perfect for college calculus or advanced high school students.

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