'underset(x to 0) (sqrt(a +x) - sqrt(a))/(x sqrt(a^(2) + ax))` is equal to

Options

A. `2 sqrt(a)`
B. `sqrt(a)`
C. `(1)/(2sqrt(a))`
D. `(1)/(2a^(3//2))`

Answer

To solve the limit problem \(\lim_{x \to 0} \frac{\sqrt{a+x} - \sqrt{a}}{x \sqrt{a^2 + ax}}\), we will follow these steps:

### Step 1: Write the limit expression
We start with the limit expression:
\[ \lim_{x \to 0} \frac{\sqrt{a+x} - \sqrt{a}}{x \sqrt{a^2 + ax}} \]

### Step 2: Rationalize the numerator
To simplify the expression, we will rationalize the numerator. We multiply and divide by the conjugate of the numerator:
\[ \frac{\sqrt{a+x} - \sqrt{a}}{x \sqrt{a^2 + ax}} \cdot \frac{\sqrt{a+x} + \sqrt{a}}{\sqrt{a+x} + \sqrt{a}} \]
This gives us:
\[ \lim_{x \to 0} \frac{(\sqrt{a+x} - \sqrt{a})(\sqrt{a+x} + \sqrt{a})}{x \sqrt{a^2 + ax} (\sqrt{a+x} + \sqrt{a})} \]
The numerator simplifies to:
\[ (\sqrt{a+x})^2 - (\sqrt{a})^2 = (a+x) - a = x \]
Thus, the expression becomes:
\[ \lim_{x \to 0} \frac{x}{x \sqrt{a^2 + ax} (\sqrt{a+x} + \sqrt{a})} \]

### Step 3: Cancel \(x\) in the numerator and denominator
Now we can cancel \(x\) from the numerator and denominator (for \(x \neq 0\)):
\[ \lim_{x \to 0} \frac{1}{\sqrt{a^2 + ax} (\sqrt{a+x} + \sqrt{a})} \]

### Step 4: Substitute \(x = 0\)
Now we substitute \(x = 0\) into the remaining expression:
\[ \sqrt{a^2 + a \cdot 0} = \sqrt{a^2} = a \]
And,
\[ \sqrt{0 + a} + \sqrt{a} = \sqrt{a} + \sqrt{a} = 2\sqrt{a} \]
Thus, we have:
\[ \lim_{x \to 0} \frac{1}{a \cdot 2\sqrt{a}} = \frac{1}{2a^{3/2}} \]

### Final Answer
The limit is:
\[ \frac{1}{2a^{3/2}} \]