The value of the limit `lim_(x to -oo) (sqrt(4x^(2) - x )+2x)` is

Options

A. `-oo`
B. `-(1)/(4)`
C. 0
D. `(1)/(4)`

Explanation

To solve the limit \( \lim_{x \to -\infty} \left( \sqrt{4x^2 - x} + 2x \right) \), we will follow these steps:

### Step 1: Analyze the expression inside the limit
We start with the expression:
\[ \sqrt{4x^2 - x} + 2x \]
As \( x \) approaches negative infinity, both \( \sqrt{4x^2 - x} \) and \( 2x \) will be influenced by the dominant term \( 4x^2 \).

### Step 2: Factor out the dominant term
We can factor out \( x^2 \) 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 negative infinity, \( |x| = -x \). Thus, we have:
\[ \sqrt{4x^2 - x} = -x\sqrt{4 - \frac{1}{x}} \]

### Step 3: Substitute back into the limit
Now substitute this back into the limit:
\[ \lim_{x \to -\infty} \left(-x\sqrt{4 - \frac{1}{x}} + 2x\right) \]
This simplifies to:
\[ \lim_{x \to -\infty} \left(-x\sqrt{4 - \frac{1}{x}} + 2x\right) = \lim_{x \to -\infty} x \left(-\sqrt{4 - \frac{1}{x}} + 2\right) \]

### Step 4: Analyze the limit of the square root
As \( x \) approaches negative infinity, \( \frac{1}{x} \) approaches 0. Therefore:
\[ \sqrt{4 - \frac{1}{x}} \to \sqrt{4} = 2 \]
Thus:
\[ -\sqrt{4 - \frac{1}{x}} + 2 \to -2 + 2 = 0 \]

### Step 5: Evaluate the limit
Now we need to evaluate:
\[ \lim_{x \to -\infty} x \cdot 0 \]
This limit approaches \( 0 \) since \( x \) is multiplied by a term that approaches \( 0 \).

### Final Answer
Thus, the value of the limit is:
\[ \boxed{0} \]