`lim_(x to a) ((cos x- cos a)/(cot x - cot a))`=

Options

A. `1/2 sin^(3)a`
B. `1/2 "cosec"^(3)a`
C. `sin^(3)a`
D. `"cosec"^(3)a `

Answer

To solve the limit \( \lim_{x \to a} \frac{\cos x - \cos a}{\cot x - \cot a} \), we will first analyze the expression.

### Step 1: Identify the limit form
When we substitute \( x = a \) directly into the expression, both the numerator and denominator become zero:
- Numerator: \( \cos a - \cos a = 0 \)
- Denominator: \( \cot a - \cot a = 0 \)

This gives us the indeterminate form \( \frac{0}{0} \), which allows us to apply L'Hôpital's rule.

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

- Derivative of the numerator \( \cos x - \cos a \):
\[ \frac{d}{dx}(\cos x) = -\sin x \]
The derivative of \( -\cos a \) is 0 since it is a constant.

- Derivative of the denominator \( \cot x - \cot a \):
\[ \frac{d}{dx}(\cot x) = -\csc^2 x \]
The derivative of \( -\cot a \) is also 0 since it is a constant.

### Step 3: Rewrite the limit using derivatives
Now we can rewrite the limit using these derivatives:
\[ \lim_{x \to a} \frac{\cos x - \cos a}{\cot x - \cot a} = \lim_{x \to a} \frac{-\sin x}{-\csc^2 x} \]
This simplifies to:
\[ \lim_{x \to a} \frac{\sin x}{\csc^2 x} = \lim_{x \to a} \sin x \cdot \sin^2 x = \lim_{x \to a} \sin^3 x \]

### Step 4: Substitute \( x = a \)
Now we can substitute \( x = a \):
\[ \sin^3 a \]

### Final Result
Thus, the limit evaluates to:
\[ \lim_{x \to a} \frac{\cos x - \cos a}{\cot x - \cot a} = \sin^3 a \]