The value of `lim_(x to oo)(d)/(dx)int_(sqrt(3))^(sqrt(x))(r^(3))/((r+1)(r-1))dr` is :

Options

A. 0
B. 1
C. `(1)/(2)`
D. None of these

Answer

To solve the given limit problem step by step, we will follow the process outlined in the video transcript.

### Step 1: Set up the limit and the integral
We start with the expression:

\[ \lim_{x \to \infty} \frac{d}{dx} \int_{\sqrt{3}}^{\sqrt{x}} \frac{r^3}{(r+1)(r-1)} \, dr \]

### Step 2: Apply Leibniz's Rule
According to Leibniz's rule for differentiation under the integral sign, we can differentiate the integral with respect to \( x \):

\[ \frac{d}{dx} \int_{a}^{b(x)} f(r) \, dr = f(b(x)) \cdot b'(x) - f(a) \cdot a'(x) \]

In our case, \( a = \sqrt{3} \) (a constant) and \( b(x) = \sqrt{x} \). Thus, we have:

\[ \frac{d}{dx} \int_{\sqrt{3}}^{\sqrt{x}} \frac{r^3}{(r+1)(r-1)} \, dr = \frac{\left(\sqrt{x}\right)^3}{(\sqrt{x}+1)(\sqrt{x}-1)} \cdot \frac{d}{dx}(\sqrt{x}) - 0 \]

### Step 3: Calculate \( b'(x) \)
The derivative of \( b(x) = \sqrt{x} \) is:

\[ b'(x) = \frac{1}{2\sqrt{x}} \]

### Step 4: Substitute into the expression
Now substituting \( b(x) \) and \( b'(x) \) into the expression gives us:

\[ = \frac{x^{3/2}}{(\sqrt{x}+1)(\sqrt{x}-1)} \cdot \frac{1}{2\sqrt{x}} \]

### Step 5: Simplify the expression
This simplifies to:

\[ = \frac{x^{3/2}}{(\sqrt{x}+1)(\sqrt{x}-1)} \cdot \frac{1}{2\sqrt{x}} = \frac{x^{3/2}}{2\sqrt{x}((\sqrt{x}+1)(\sqrt{x}-1))} = \frac{x^{3/2}}{2\sqrt{x}(x-1)} = \frac{x}{2(x-1)} \]

### Step 6: Take the limit as \( x \to \infty \)
Now we take the limit:

\[ \lim_{x \to \infty} \frac{x}{2(x-1)} = \lim_{x \to \infty} \frac{x}{2x - 2} = \lim_{x \to \infty} \frac{1}{2 - \frac{2}{x}} = \frac{1}{2} \]

### Final Answer
Thus, the value of the limit is:

\[ \frac{1}{2} \]