If `lim_(xrarr0) (x(1+acos x)-bsin x)/(x)=1`, then a-b, are

Options

A. `1/2
B. `5/2
C. `-5/2
D. 0

Answer

To solve the limit problem given by

\[ \lim_{x \to 0} \frac{x(1 + a \cos x) - b \sin x}{x} = 1, \]

we will follow these steps:

### Step 1: Simplify the limit expression
We start with the limit expression:

\[ \lim_{x \to 0} \frac{x(1 + a \cos x) - b \sin x}{x}. \]

We can separate the terms in the numerator:

\[ \lim_{x \to 0} \left( (1 + a \cos x) - \frac{b \sin x}{x} \right). \]

### Step 2: Apply the limit properties
Now we can evaluate the limit of each part separately. We know that:

1. \(\lim_{x \to 0} \cos x = 1\)
2. \(\lim_{x \to 0} \frac{\sin x}{x} = 1\)

Thus, we have:

\[ \lim_{x \to 0} (1 + a \cos x) = 1 + a \cdot 1 = 1 + a, \]

and

\[ \lim_{x \to 0} \frac{b \sin x}{x} = b \cdot 1 = b. \]

### Step 3: Set up the equation
Now substituting these limits back into our expression, we get:

\[ 1 + a - b = 1. \]

### Step 4: Solve for \(a - b\)
From the equation \(1 + a - b = 1\), we can simplify it:

\[ a - b = 1 - 1 = 0. \]

### Final Result
Thus, we find that:

\[ a - b = 0. \]