`lim_(x->a)(x)/(x-a)int_(a)^(x)f(x)dx` equals

Options

A. `2f(a)`
B. f(a)
C. af(a)
D. 0

Answer

To solve the limit \( \lim_{x \to a} \frac{x}{x-a} \int_{a}^{x} f(t) \, dt \), we will follow these steps:

### Step 1: Identify the form of the limit
As \( x \to a \), the integral \( \int_{a}^{x} f(t) \, dt \) approaches \( 0 \) because the limits of integration are the same. Thus, we have:
\[ \lim_{x \to a} \frac{x}{x-a} \int_{a}^{x} f(t) \, dt = \frac{a}{0} \cdot 0 = \frac{0}{0} \]
This is an indeterminate form \( \frac{0}{0} \).

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule. This involves differentiating the numerator and denominator with respect to \( x \).

### Step 3: Differentiate the numerator and denominator
The numerator is \( \int_{a}^{x} f(t) \, dt \) and the denominator is \( x - a \).

Using the Fundamental Theorem of Calculus, the derivative of the numerator is:
\[ \frac{d}{dx} \left( \int_{a}^{x} f(t) \, dt \right) = f(x) \]
The derivative of the denominator is:
\[ \frac{d}{dx} (x - a) = 1 \]

### Step 4: Rewrite the limit
Now we can rewrite the limit using the derivatives:
\[ \lim_{x \to a} \frac{f(x)}{1} = f(a) \]

### Step 5: Substitute back into the limit
Thus, we have:
\[ \lim_{x \to a} \frac{x}{x-a} \int_{a}^{x} f(t) \, dt = f(a) \]

### Final Result
The final result is:
\[ \lim_{x \to a} \frac{x}{x-a} \int_{a}^{x} f(t) \, dt = a \cdot f(a) \]