Evaluate`lim_(x to a) (sqrt(4a+3x)-sqrt(x+6a))/(sqrt(2a+5x)-sqrt(3a+4x))`

Answer

To evaluate the limit

\[ \lim_{x \to a} \frac{\sqrt{4a + 3x} - \sqrt{x + 6a}}{\sqrt{2a + 5x} - \sqrt{3a + 4x}}, \]

we will use the technique of multiplying by the conjugate.

### Step 1: Multiply by the Conjugate
We multiply the numerator and denominator by the conjugate of the numerator:

\[ \frac{\sqrt{4a + 3x} - \sqrt{x + 6a}}{\sqrt{2a + 5x} - \sqrt{3a + 4x}} \cdot \frac{\sqrt{4a + 3x} + \sqrt{x + 6a}}{\sqrt{4a + 3x} + \sqrt{x + 6a}}. \]

This gives us:

\[ \frac{(4a + 3x) - (x + 6a)}{(\sqrt{2a + 5x} - \sqrt{3a + 4x})(\sqrt{4a + 3x} + \sqrt{x + 6a})}. \]

### Step 2: Simplify the Numerator
Now simplify the numerator:

\[ (4a + 3x) - (x + 6a) = 4a + 3x - x - 6a = -2a + 2x = 2(x - a). \]

### Step 3: Rewrite the Limit
Now substitute this back into the limit:

\[ \lim_{x \to a} \frac{2(x - a)}{(\sqrt{2a + 5x} - \sqrt{3a + 4x})(\sqrt{4a + 3x} + \sqrt{x + 6a})}. \]

### Step 4: Multiply by the Conjugate of the Denominator
Next, we multiply the denominator by its conjugate:

\[ \frac{\sqrt{2a + 5x} + \sqrt{3a + 4x}}{\sqrt{2a + 5x} + \sqrt{3a + 4x}}. \]

This gives us:

\[ \frac{2(x - a)(\sqrt{2a + 5x} + \sqrt{3a + 4x})}{(2a + 5x) - (3a + 4x)}. \]

### Step 5: Simplify the Denominator
Now simplify the denominator:

\[ (2a + 5x) - (3a + 4x) = 2a + 5x - 3a - 4x = -a + x. \]

### Step 6: Rewrite the Limit Again
Now we have:

\[ \lim_{x \to a} \frac{2(x - a)(\sqrt{2a + 5x} + \sqrt{3a + 4x})}{x - a}. \]

### Step 7: Cancel the Common Factor
We can cancel \(x - a\) (since we are taking the limit as \(x\) approaches \(a\), we can safely cancel it):

\[ \lim_{x \to a} 2(\sqrt{2a + 5x} + \sqrt{3a + 4x}). \]

### Step 8: Substitute \(x = a\)
Now substitute \(x = a\):

\[ 2(\sqrt{2a + 5a} + \sqrt{3a + 4a}) = 2(\sqrt{7a} + \sqrt{7a}) = 2(2\sqrt{7a}) = 4\sqrt{7a}. \]

### Final Answer
Thus, the limit evaluates to:

\[ \boxed{4\sqrt{7a}}. \]