If `f(x) = {(sqrt(1 + sqrt(5 + x)-a)/((x-4)), 0 le x le 4),(b, x ge 4):}` is continuous at x = 4 then value of `1/(ab)` is equal to

Options

A. 4
B. 15
C. 8
D. 12

Answer

To solve the problem, we need to ensure that the function \( f(x) \) is continuous at \( x = 4 \). This means that the left-hand limit (LHL) and right-hand limit (RHL) at \( x = 4 \) must be equal, and they must also equal the value of the function at that point.

### Step-by-Step Solution:

1. **Identify the Function**:
The function is defined as:
\[ f(x) = \begin{cases} \frac{\sqrt{1 + \sqrt{5 + x}} - a}{x - 4} & \text{for } 0 \leq x < 4 \\ b & \text{for } x \geq 4 \end{cases} \]

2. **Find the Left-Hand Limit (LHL) as \( x \) approaches 4**:
We need to calculate:
\[ \lim_{x \to 4^-} f(x) = \lim_{x \to 4} \frac{\sqrt{1 + \sqrt{5 + x}} - a}{x - 4} \]
Substituting \( x = 4 \):
\[ \sqrt{5 + 4} = \sqrt{9} = 3 \implies \sqrt{1 + 3} = \sqrt{4} = 2 \]
Thus, we have:
\[ \lim_{x \to 4} \frac{2 - a}{x - 4} \]
This is an indeterminate form \( \frac{0}{0} \) when \( a = 2 \). Therefore, we set \( a = 2 \).

3. **Apply L'Hôpital's Rule**:
Since we have an indeterminate form, we can apply L'Hôpital's Rule:
\[ \lim_{x \to 4} \frac{2 - a}{x - 4} = \lim_{x \to 4} \frac{\frac{d}{dx}(\sqrt{1 + \sqrt{5 + x}} - a)}{\frac{d}{dx}(x - 4)} \]
The derivative of the numerator:
\[ \frac{d}{dx}(\sqrt{1 + \sqrt{5 + x}}) = \frac{1}{2\sqrt{1 + \sqrt{5 + x}}} \cdot \frac{1}{2\sqrt{5 + x}} = \frac{1}{4\sqrt{1 + \sqrt{5 + x}} \sqrt{5 + x}} \]
The derivative of the denominator is simply \( 1 \).

4. **Evaluate the Limit**:
Now substituting \( x = 4 \):
\[ \lim_{x \to 4} \frac{1}{4\sqrt{1 + 3}\sqrt{9}} = \frac{1}{4 \cdot 2 \cdot 3} = \frac{1}{24} \]
Thus, \( \text{LHL} = \frac{1}{24} \).

5. **Find the Right-Hand Limit (RHL)**:
The RHL at \( x = 4 \) is simply \( f(4) = b \).

6. **Set the Limits Equal for Continuity**:
For continuity at \( x = 4 \):
\[ \frac{1}{24} = b \]

7. **Calculate \( 1/(ab) \)**:
We have \( a = 2 \) and \( b = \frac{1}{24} \):
\[ 1/(ab) = 1/(2 \cdot \frac{1}{24}) = 1/( \frac{2}{24}) = 1/( \frac{1}{12}) = 12 \]

### Final Answer:
\[ \frac{1}{ab} = 12 \]