`lim_(x to 2a) (sqrt(x-2a) + sqrt(x)- sqrt(2a))/sqrt(x^(2) - 4a^(2))=`

Options

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

Answer

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

### Step 1: Substitute \( x = 2a \)
First, we substitute \( x = 2a \) directly into the expression to check if we get an indeterminate form:
\[ \sqrt{2a - 2a} + \sqrt{2a} - \sqrt{2a} = 0 + \sqrt{2a} - \sqrt{2a} = 0 \]
For the denominator:
\[ \sqrt{(2a)^2 - 4a^2} = \sqrt{4a^2 - 4a^2} = \sqrt{0} = 0 \]
Thus, we have the indeterminate form \( \frac{0}{0} \).

### Step 2: Apply L'Hôpital's Rule
Since we have the indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule, which states that:
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]
where \( f(x) = \sqrt{x - 2a} + \sqrt{x} - \sqrt{2a} \) and \( g(x) = \sqrt{x^2 - 4a^2} \).

### Step 3: Differentiate the Numerator and Denominator
Now we differentiate the numerator and the denominator.

**Numerator:**
1. The derivative of \( \sqrt{x - 2a} \) is \( \frac{1}{2\sqrt{x - 2a}} \).
2. The derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \).
3. The derivative of \( \sqrt{2a} \) is \( 0 \).

Thus, the derivative of the numerator is:
\[ f'(x) = \frac{1}{2\sqrt{x - 2a}} + \frac{1}{2\sqrt{x}} \]

**Denominator:**
The derivative of \( \sqrt{x^2 - 4a^2} \) is:
\[ g'(x) = \frac{1}{2\sqrt{x^2 - 4a^2}} \cdot 2x = \frac{x}{\sqrt{x^2 - 4a^2}} \]

### Step 4: Rewrite the Limit
Now we rewrite the limit using the derivatives:
\[ \lim_{x \to 2a} \frac{\frac{1}{2\sqrt{x - 2a}} + \frac{1}{2\sqrt{x}}}{\frac{x}{\sqrt{x^2 - 4a^2}}} \]

### Step 5: Simplify the Expression
We can simplify this expression:
\[ = \lim_{x \to 2a} \frac{\frac{1}{2\sqrt{x - 2a}} + \frac{1}{2\sqrt{x}}}{\frac{x}{\sqrt{x^2 - 4a^2}}} = \lim_{x \to 2a} \frac{\sqrt{x^2 - 4a^2}}{x} \left( \frac{1}{2\sqrt{x - 2a}} + \frac{1}{2\sqrt{x}} \right) \]

### Step 6: Substitute \( x = 2a \) Again
Now we substitute \( x = 2a \):
1. The numerator becomes:
\[ \sqrt{(2a)^2 - 4a^2} = \sqrt{0} = 0 \]
2. The denominator becomes:
\[ 2a \]
Thus, we need to evaluate:
\[ \lim_{x \to 2a} \left( \frac{1}{2\sqrt{x - 2a}} + \frac{1}{2\sqrt{x}} \right) \]
The first term approaches infinity as \( x \to 2a \), while the second term approaches \( \frac{1}{2\sqrt{2a}} \).

### Step 7: Final Limit Calculation
Now we can evaluate the limit:
\[ \lim_{x \to 2a} \frac{\sqrt{x - 2a} + \sqrt{x}}{\sqrt{x^2 - 4a^2}} = \frac{\sqrt{2a} + 0}{2a} = \frac{\sqrt{2a}}{2a} = \frac{1}{2\sqrt{a}} \]

### Final Answer
Thus, the final answer is:
\[ \frac{1}{2\sqrt{a}} \]