If `f(x)={{:((sqrtx)/(sqrt(4+sqrtx)-a),xgt0),(c,x=0),((4e^((2)/(x))+3e^((1)/(x)))/(e^((2)/(x))+be^((1)/(x))),xlt0):}`
continuous at x = 0 for some constants a, b and c, then the value of `(50b)/(a)` is equal to

Answer

To solve the problem, we need to ensure that the function \( f(x) \) is continuous at \( x = 0 \). This means that the right-hand limit (RHL), the left-hand limit (LHL), and the functional value at \( x = 0 \) must all be equal.

### Step-by-step Solution:

1. **Identify the function pieces**:
The function \( f(x) \) is defined as:
\[ f(x) = \begin{cases} \frac{\sqrt{x}}{\sqrt{4 + \sqrt{x}} - a} & \text{if } x > 0 \\ c & \text{if } x = 0 \\ \frac{4e^{\frac{2}{x}} + 3e^{\frac{1}{x}}}{e^{\frac{2}{x}} + be^{\frac{1}{x}}} & \text{if } x < 0 \end{cases} \]

2. **Calculate the Right-Hand Limit (RHL)**:
For \( x > 0 \):
\[ \text{RHL} = \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \frac{\sqrt{x}}{\sqrt{4 + \sqrt{x}} - a} \]
Substitute \( t = \sqrt{x} \) (thus \( x = t^2 \)):
\[ \text{RHL} = \lim_{t \to 0} \frac{t}{\sqrt{4 + t} - a} \]
This limit will be in the form \( \frac{0}{0} \) if \( \sqrt{4} - a = 0 \), which gives \( a = 2 \).

3. **Substituting \( a \) into the limit**:
Now substituting \( a = 2 \):
\[ \text{RHL} = \lim_{t \to 0} \frac{t}{\sqrt{4 + t} - 2} \]
Rationalizing the denominator:
\[ = \lim_{t \to 0} \frac{t(\sqrt{4 + t} + 2)}{(4 + t) - 4} = \lim_{t \to 0} \frac{t(\sqrt{4 + t} + 2)}{t} = \lim_{t \to 0} (\sqrt{4 + t} + 2) \]
As \( t \to 0 \):
\[ \text{RHL} = 2 + 2 = 4 \]

4. **Calculate the Left-Hand Limit (LHL)**:
For \( x < 0 \):
\[ \text{LHL} = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{4e^{\frac{2}{x}} + 3e^{\frac{1}{x}}}{e^{\frac{2}{x}} + be^{\frac{1}{x}}} \]
As \( x \to 0^- \), \( e^{\frac{1}{x}} \to 0 \) and \( e^{\frac{2}{x}} \to 0 \):
\[ \text{LHL} = \frac{0 + 0}{0 + b} = 0 \quad \text{(if } b \neq 0\text{)} \]

5. **Equate limits and functional value**:
Since \( f(0) = c \), we have:
\[ \text{RHL} = c = 4 \]
Thus, \( c = 4 \).

6. **Equate LHL to \( c \)**:
From the left-hand limit:
\[ \frac{3}{b} = 4 \implies b = \frac{3}{4} \]

7. **Calculate \( \frac{50b}{a} \)**:
Now, we have \( a = 2 \) and \( b = \frac{3}{4} \):
\[ \frac{50b}{a} = \frac{50 \cdot \frac{3}{4}}{2} = \frac{50 \cdot 3}{8} = \frac{150}{8} = 18.75 \]

### Final Answer:
\[ \frac{50b}{a} = 18.75 \]