The value of `lim_(x to infty) sqrt(a^(2)x^(2)+ax +1) - sqrt(a^(2)x^(2))` is:

Options

A. `1//2`
B. 1
C. 2
D. none of these

Answer

To solve the limit problem, we need to evaluate the expression:

\[ \lim_{x \to \infty} \left( \sqrt{a^2 x^2 + ax + 1} - \sqrt{a^2 x^2} \right) \]

### Step 1: Rewrite the limit expression
We can rewrite the limit as:

\[ L = \lim_{x \to \infty} \left( \sqrt{a^2 x^2 + ax + 1} - \sqrt{a^2 x^2} \right) \]

### Step 2: Factor out \(x^2\) from the square roots
Notice that both square roots contain \(x^2\). We can factor \(x^2\) out of the square root:

\[ L = \lim_{x \to \infty} \left( \sqrt{x^2 \left( a^2 + \frac{a}{x} + \frac{1}{x^2} \right)} - \sqrt{x^2 a^2} \right) \]

This simplifies to:

\[ L = \lim_{x \to \infty} \left( x \sqrt{a^2 + \frac{a}{x} + \frac{1}{x^2}} - x \sqrt{a^2} \right) \]

### Step 3: Factor out \(x\)
Now we can factor \(x\) out of the limit:

\[ L = \lim_{x \to \infty} x \left( \sqrt{a^2 + \frac{a}{x} + \frac{1}{x^2}} - \sqrt{a^2} \right) \]

### Step 4: Simplify the expression inside the limit
Next, we simplify the expression inside the parentheses. We can use the difference of squares:

\[ L = \lim_{x \to \infty} x \cdot \frac{\left( a^2 + \frac{a}{x} + \frac{1}{x^2} \right) - a^2}{\sqrt{a^2 + \frac{a}{x} + \frac{1}{x^2}} + \sqrt{a^2}} \]

This simplifies to:

\[ L = \lim_{x \to \infty} x \cdot \frac{\frac{a}{x} + \frac{1}{x^2}}{\sqrt{a^2 + \frac{a}{x} + \frac{1}{x^2}} + \sqrt{a^2}} \]

### Step 5: Simplify further
Now we can simplify the numerator:

\[ L = \lim_{x \to \infty} \frac{a + \frac{1}{x}}{\sqrt{a^2 + \frac{a}{x} + \frac{1}{x^2}} + \sqrt{a^2}} \]

### Step 6: Evaluate the limit
As \(x \to \infty\), \(\frac{1}{x} \to 0\), so we have:

\[ L = \frac{a}{\sqrt{a^2} + \sqrt{a^2}} = \frac{a}{2\sqrt{a^2}} = \frac{a}{2|a|} \]

### Step 7: Final result
Thus, the limit evaluates to:

\[ L = \frac{1}{2} \quad \text{(if } a > 0\text{)} \]