Let function f be defined as `f:R^(+)toR^(+)` and function g is defined as `g:R^(+)toR^(+)`. Functions f and g are continuous in their domain. Suppose, the function `h(x)=lim_(ntooo)(x^(n)f(x)+x^(2))/(x^(n)+g(x)), x gt0`. If `h(x)` is continuous in its domain,then`f(1).g(1)` is equal to (a) 2 (b) 1 (c) `1/2` (d) 0

Options

A. 2
B. 1
C. `1/2`
D. 0

Answer

To solve the problem step by step, we will analyze the limit function \( h(x) \) and its continuity.

### Step 1: Write down the function \( h(x) \)
We have the function defined as:
\[ h(x) = \lim_{n \to \infty} \frac{x^n f(x) + x^2}{x^n + g(x)} \]

### Step 2: Factor out \( x^n \) from the numerator and denominator
To simplify \( h(x) \), we factor \( x^n \) out of both the numerator and the denominator:
\[ h(x) = \lim_{n \to \infty} \frac{x^n (f(x) + \frac{x^2}{x^n})}{x^n (1 + \frac{g(x)}{x^n})} \]
This simplifies to:
\[ h(x) = \lim_{n \to \infty} \frac{f(x) + \frac{x^2}{x^n}}{1 + \frac{g(x)}{x^n}} \]

### Step 3: Evaluate the limit as \( n \to \infty \)
As \( n \to \infty \), \( \frac{x^2}{x^n} \) approaches 0 (since \( x > 0 \)) and \( \frac{g(x)}{x^n} \) also approaches 0. Thus, we have:
\[ h(x) = \frac{f(x) + 0}{1 + 0} = f(x) \]

### Step 4: Analyze the continuity of \( h(x) \)
Since \( h(x) = f(x) \) and both functions are continuous in their domain, \( h(x) \) is also continuous.

### Step 5: Evaluate \( h(1) \)
Now, we evaluate \( h(1) \):
\[ h(1) = f(1) \]

### Step 6: Evaluate \( h(x) \cdot g(x) \)
From the earlier steps, we know:
\[ h(x) \cdot g(x) = f(x) \cdot g(x) \]
Since \( h(x) \) is continuous and equals \( f(x) \), we can write:
\[ f(x) \cdot g(x) = 1 \]

### Step 7: Evaluate at \( x = 1 \)
Substituting \( x = 1 \):
\[ f(1) \cdot g(1) = 1 \]

### Conclusion
Thus, the value of \( f(1) \cdot g(1) \) is equal to 1.

### Final Answer
The answer is (b) 1.