Let `g (x ) = x ^(C )e ^(Cx) and f (x) = int _(0)^(x) te ^(2r) (1+3t ^(2))^(1//2) dt. If L = lim _(x to oo) (f'(x ))/(g '(x))` is non-zero finite number then :
The valur of C is:

Options

A. 7
B. `3/2`
C. `2`
D. 3

Answer

To solve the problem, we need to find the value of \( C \) such that the limit

\[ L = \lim_{x \to \infty} \frac{f'(x)}{g'(x)} \]

is a non-zero finite number.

### Step 1: Define the functions
We have:

\[ g(x) = x^C e^{Cx} \]

and

\[ f(x) = \int_0^x t e^{2t} \sqrt{1 + 3t^2} \, dt \]

### Step 2: Differentiate \( g(x) \)
Using the product rule, we differentiate \( g(x) \):

\[ g'(x) = \frac{d}{dx}(x^C e^{Cx}) = C x^{C-1} e^{Cx} + x^C \cdot C e^{Cx} = C e^{Cx} (x^{C-1} + x^C) = C e^{Cx} x^{C-1}(1 + x) \]

### Step 3: Differentiate \( f(x) \)
Using the Fundamental Theorem of Calculus, we differentiate \( f(x) \):

\[ f'(x) = x e^{2x} \sqrt{1 + 3x^2} \]

### Step 4: Set up the limit
Now we set up the limit:

\[ L = \lim_{x \to \infty} \frac{f'(x)}{g'(x)} = \lim_{x \to \infty} \frac{x e^{2x} \sqrt{1 + 3x^2}}{C e^{Cx} x^{C-1}(1 + x)} \]

### Step 5: Simplify the limit
We can simplify the limit:

\[ L = \lim_{x \to \infty} \frac{x e^{2x} \sqrt{1 + 3x^2}}{C e^{Cx} x^{C-1}(1 + x)} = \lim_{x \to \infty} \frac{e^{2x} \sqrt{1 + 3x^2}}{C e^{Cx} x^{C-2}(1 + x)} \]

### Step 6: Analyze the limit
As \( x \to \infty \), we can approximate:

\[ \sqrt{1 + 3x^2} \approx \sqrt{3} x \]

Thus,

\[ L = \lim_{x \to \infty} \frac{e^{2x} \sqrt{3} x}{C e^{Cx} x^{C-2}(1 + x)} = \lim_{x \to \infty} \frac{\sqrt{3} x e^{2x}}{C e^{Cx} x^{C-2}(1 + x)} \]

### Step 7: Further simplification
This can be simplified to:

\[ L = \lim_{x \to \infty} \frac{\sqrt{3} x e^{2x}}{C e^{Cx} x^{C-2}(x)} = \lim_{x \to \infty} \frac{\sqrt{3} e^{2x}}{C e^{Cx} x^{C-1}} \]

### Step 8: Determine conditions for a finite limit
For \( L \) to be finite and non-zero, the exponent of \( e \) in the numerator must equal the exponent in the denominator, which gives:

\[ 2 - C = 0 \implies C = 2 \]

### Conclusion
Thus, the value of \( C \) is:

\[ \boxed{2} \]