In a `Delta ABC`, angles A, B, C are in AP. If `f(x) = lim_(A to c) (sqrt(3 - 4 sin A sin C))/(|A-C|)`, then f'(x) is equal to ..........

Answer

To solve the problem step by step, we will analyze the given information and apply the necessary mathematical concepts.

### Step 1: Understand the Angles in Triangle ABC
Given that angles A, B, and C are in Arithmetic Progression (AP), we can express them as:
- Let A = a
- Let B = b
- Let C = c

Since they are in AP, we have:
\[ 2B = A + C \]
This implies:
\[ B = \frac{A + C}{2} \]

### Step 2: Use the Sum of Angles in a Triangle
We know that the sum of angles in a triangle is:
\[ A + B + C = \pi \]
Substituting \( B \) from the previous step:
\[ A + \frac{A + C}{2} + C = \pi \]

### Step 3: Simplify the Equation
Multiplying through by 2 to eliminate the fraction:
\[ 2A + A + C + 2C = 2\pi \]
This simplifies to:
\[ 3A + 3C = 2\pi \]
Thus:
\[ A + C = \frac{2\pi}{3} \]

### Step 4: Substitute into the Limit Function
Now we need to evaluate:
\[ f(x) = \lim_{A \to C} \frac{\sqrt{3 - 4 \sin A \sin C}}{|A - C|} \]

Using the identity:
\[ \sin A \sin C = \frac{1}{2} (\cos(A - C) - \cos(A + C)) \]
We can express:
\[ 4 \sin A \sin C = 2 (\cos(A - C) - \cos(A + C)) \]

### Step 5: Substitute Values
Now substituting \( A + C = \frac{2\pi}{3} \):
\[ \cos(A + C) = \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \]
Thus:
\[ 4 \sin A \sin C = 2 (\cos(A - C) + \frac{1}{2}) \]

### Step 6: Simplify the Expression
Now substituting back into the limit:
\[ f(x) = \lim_{A \to C} \frac{\sqrt{3 - 2(\cos(A - C) + \frac{1}{2})}}{|A - C|} \]
This simplifies to:
\[ \sqrt{3 - 2\cos(A - C) - 1} = \sqrt{2 - 2\cos(A - C)} = \sqrt{2(1 - \cos(A - C)} = \sqrt{4\sin^2\left(\frac{A - C}{2}\right)} = 2|\sin\left(\frac{A - C}{2}\right)| \]

### Step 7: Evaluate the Limit
Thus:
\[ f(x) = \lim_{A \to C} \frac{2|\sin\left(\frac{A - C}{2}\right)|}{|A - C|} \]
Using the property:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
We can rewrite:
\[ f(x) = \lim_{A \to C} \frac{2|\sin\left(\frac{A - C}{2}\right)|}{|A - C|} = \lim_{A \to C} \frac{2|\sin\left(\frac{A - C}{2}\right)|}{2|A - C|/2} = 1 \]

### Step 8: Differentiate f(x)
Since \( f(x) = 1 \), we find:
\[ f'(x) = 0 \]

### Final Answer
Thus, \( f'(x) \) is equal to:
\[ \boxed{0} \]