Find the limit. (*Hint:* Treat the expression as a fraction whose denominator is 1, and rationalize the numerator.) Use a graphing utility to verify your result.

 lim *x*→−∞ 2*x* + sqrt(4*x*2 − *x)*
 

To find the limit as \( x \) approaches \(-\infty\) for the given expression:

\[
\lim_{x \to -\infty} \left( 2x + \sqrt{4x^2 - x} \right)
\]

we can follow the hint and treat the expression as a fraction with a denominator of 1, then rationalize the numerator.

### Step 1: Original Expression
The expression is:

\[
2x + \sqrt{4x^2 - x}
\]

### Step 2: Rationalize the Expression
To rationalize the expression, we multiply the numerator and the "denominator" (which is implicitly 1 here) by the conjugate of the expression \( 2x + \sqrt{4x^2 - x} \). The conjugate is:

\[
2x - \sqrt{4x^2 - x}
\]

Thus, the expression becomes:

\[
\lim_{x \to -\infty} \frac{\left( 2x + \sqrt{4x^2 - x} \right) \left( 2x - \sqrt{4x^2 - x} \right)}{2x - \sqrt{4x^2 - x}}
\]

### Step 3: Simplify the Numerator
Using the difference of squares formula:

\[
(a + b)(a - b) = a^2 - b^2
\]

we have:

\[
(2x)^2 - \left(\sqrt{4x^2 - x}\right)^2 = 4x^2 - (4x^2 - x) = x
\]

So the expression simplifies to:

\[
\lim_{x \to -\infty} \frac{x}{2x - \sqrt{4x^2 - x}}
\]

### Step 4: Factor and Simplify Further
We can factor out \( x \) from the square root:

\[
\sqrt{4x^2 - x} = \sqrt{x^2(4 - \frac{1}{x})} = |x|\sqrt{4 - \frac{1}{x}}
\]

Since \( x \) is approaching \(-\infty\), \( |x| = -x \). Therefore:

\[
\sqrt{4x^2 - x} = -x\sqrt{4 - \frac{1}{x}}
\]

Substitute this back into the expression:

\[
\lim_{x \to -\infty} \frac{x}{2x - (-x)\sqrt{4 - \frac{1}{x}}}
\]

This simplifies to:

\[
\lim_{x \to -\infty} \frac{x}{x(2 + \sqrt{4 - \frac{1}{x}})}
\]

\[
\lim_{x \to -\infty} \frac{1}{2 + \sqrt{4 - \frac{1}{x}}}
\]

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

\[
\lim_{x \to -\infty} \frac{1}{2 + \sqrt{4}} = \frac{1}{2 + 2} = \frac{1}{4}
\]

Thus, the limit is:

\[
\boxed{\frac{1}{4}}
\]

### Verification Using a Graphing Utility
You can use a graphing calculator or software to plot the function \( f(x) = 2x + \sqrt{4x^2 - x} \) as \( x \) approaches negative infinity and confirm that the value of the expression approaches \(\frac{1}{4}\).

Would you like more details on using graphing utilities, or do you have any other questions?

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### Follow-up Questions:

1. How does the limit change if \( x \) approaches \( +\infty \)?
2. Why is rationalizing the expression useful in finding the limit?
3. Can the same method be used to find the limit for other similar expressions?
4. How does the behavior of the square root function influence the limit?
5. What happens if we do not rationalize and directly substitute \( x \) as \( -\infty \)?

### Tip:
Rationalizing the numerator or denominator can be a powerful technique for evaluating limits, especially when dealing with square roots or complex expressions.

Learn how to find the limit of the expression 2x + sqrt(4x^2 - x) as x approaches negative infinity using rationalization techniques. Step-by-step solution provided with the use of difference of squares formula and verification using a graphing utility.

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