Let `f(x)=int_(-x)^(x)(t sin at + bt +c) dt, ` where a,b,c are non-zero real numbers , then `lim_(x rarr0) (f(x))/(x)` is

Options

A. independent of a
B. independent of a and b , and has the value equals to c
C. independent a, b and c
D. dependent only on c

Answer

To solve the problem, we need to find the limit:

\[ \lim_{x \to 0} \frac{f(x)}{x} \]

where

\[ f(x) = \int_{-x}^{x} (t \sin(at) + bt + c) \, dt \]

### Step 1: Express the limit

We start by substituting the expression for \( f(x) \):

\[ \lim_{x \to 0} \frac{f(x)}{x} = \lim_{x \to 0} \frac{1}{x} \int_{-x}^{x} (t \sin(at) + bt + c) \, dt \]

### Step 2: Evaluate the integral

We can split the integral into three parts:

\[ \int_{-x}^{x} (t \sin(at)) \, dt + \int_{-x}^{x} (bt) \, dt + \int_{-x}^{x} c \, dt \]

1. **First Integral: \( \int_{-x}^{x} t \sin(at) \, dt \)**

This integral is an odd function (since \( t \sin(at) \) is odd), so:

\[ \int_{-x}^{x} t \sin(at) \, dt = 0 \]

2. **Second Integral: \( \int_{-x}^{x} bt \, dt \)**

This is also an odd function, so:

\[ \int_{-x}^{x} bt \, dt = 0 \]

3. **Third Integral: \( \int_{-x}^{x} c \, dt \)**

This integral is constant:

\[ \int_{-x}^{x} c \, dt = c \cdot (x - (-x)) = c \cdot 2x = 2cx \]

### Step 3: Combine the results

Now we can combine the results of the integrals:

\[ f(x) = 0 + 0 + 2cx = 2cx \]

### Step 4: Substitute back into the limit

Now substitute \( f(x) \) back into the limit:

\[ \lim_{x \to 0} \frac{f(x)}{x} = \lim_{x \to 0} \frac{2cx}{x} \]

### Step 5: Simplify the limit

This simplifies to:

\[ \lim_{x \to 0} 2c = 2c \]

### Final Answer

Thus, the limit is:

\[ \lim_{x \to 0} \frac{f(x)}{x} = 2c \]