`lim_(xrarr0) (int_(0)^(x)(t^(2))/(sqrt(a+t))dt)/(x-sinx)=1(agt0)`. Then the value of a is

Options

A. `1//2`
B. `1//4`
C. 2
D. 4

Explanation

To solve the limit problem given, we need to evaluate the following expression:

\[ \lim_{x \to 0} \frac{\int_0^x \frac{t^2}{\sqrt{a+t}} dt}{x - \sin x} = 1 \quad (a > 0) \]

### Step 1: Identify the Form of the Limit
As \( x \to 0 \), both the numerator and denominator approach 0. Thus, we have the indeterminate form \( \frac{0}{0} \).

**Hint:** Recognize when to apply L'Hôpital's Rule, which is useful for indeterminate forms.

### Step 2: Apply L'Hôpital's Rule
According to L'Hôpital's Rule, we differentiate the numerator and the denominator separately.

**Numerator:**
\[ \text{Let } I(x) = \int_0^x \frac{t^2}{\sqrt{a+t}} dt \]
Using the Fundamental Theorem of Calculus, we differentiate:
\[ I'(x) = \frac{x^2}{\sqrt{a+x}} \]

**Denominator:**
\[ \frac{d}{dx}(x - \sin x) = 1 - \cos x \]

**Hint:** Remember to differentiate both the numerator and the denominator.

### Step 3: Rewrite the Limit
Now we can rewrite the limit using the derivatives:
\[ \lim_{x \to 0} \frac{\frac{x^2}{\sqrt{a+x}}}{1 - \cos x} \]

### Step 4: Substitute \( x = 0 \) Again
As \( x \to 0 \), we again encounter the form \( \frac{0}{0} \). We apply L'Hôpital's Rule again.

**Numerator:**
\[ \frac{d}{dx}\left(\frac{x^2}{\sqrt{a+x}}\right) = \frac{2x \sqrt{a+x} - \frac{1}{2} x^2 \cdot \frac{1}{\sqrt{a+x}}}{a+x} \]
This simplifies to:
\[ \frac{2x(a+x) - \frac{1}{2} x^2}{(a+x)\sqrt{a+x}} \]

**Denominator:**
\[ \frac{d}{dx}(1 - \cos x) = \sin x \]

**Hint:** Differentiate carefully, and simplify where possible.

### Step 5: Rewrite the Limit Again
Now we have:
\[ \lim_{x \to 0} \frac{\frac{2x(a+x) - \frac{1}{2} x^2}{(a+x)\sqrt{a+x}}}{\sin x} \]

### Step 6: Use the Limit of \( \sin x \)
We can rewrite \( \sin x \) as \( x \cdot \frac{\sin x}{x} \) and use the fact that \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \):
\[ \lim_{x \to 0} \frac{2(a+x) - \frac{1}{2} x}{(a+x)\sqrt{a+x}} = \frac{2a}{a\sqrt{a}} = \frac{2}{\sqrt{a}} \]

### Step 7: Set the Limit Equal to 1
According to the problem statement, this limit equals 1:
\[ \frac{2}{\sqrt{a}} = 1 \]

### Step 8: Solve for \( a \)
Squaring both sides gives:
\[ 4 = a \quad \Rightarrow \quad a = 4 \]

### Final Answer
The value of \( a \) is:
\[ \boxed{4} \]