The value of `lim _(x to 0) [(a)/(x) - cot ""(x)/(a)]` is

Answer

To solve the limit \( \lim_{x \to 0} \left( \frac{a}{x} - \frac{\cot(x)}{a} \right) \), we will follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the limit expression:
\[ \lim_{x \to 0} \left( \frac{a}{x} - \frac{\cot(x)}{a} \right) \]
This can be expressed with a common denominator:
\[ = \lim_{x \to 0} \left( \frac{a^2 - x \cos(x)}{ax} \right) \]

### Step 2: Simplify the limit
Now, we can rewrite the limit as:
\[ = \lim_{x \to 0} \frac{a^2 - x \cos(x)}{ax} \]
We can factor out \( a \) from the denominator:
\[ = \frac{1}{a} \lim_{x \to 0} \frac{a^2 - x \cos(x)}{x} \]

### Step 3: Evaluate the limit
As \( x \to 0 \), both the numerator and denominator approach 0, resulting in the indeterminate form \( \frac{0}{0} \). We can apply L'Hôpital's Rule:
\[ \text{Differentiate the numerator: } \frac{d}{dx}(a^2 - x \cos(x)) = -\cos(x) + x \sin(x) \]
\[ \text{Differentiate the denominator: } \frac{d}{dx}(x) = 1 \]
Thus, we can rewrite the limit:
\[ = \frac{1}{a} \lim_{x \to 0} \frac{-\cos(x) + x \sin(x)}{1} \]

### Step 4: Substitute \( x = 0 \)
Now, we substitute \( x = 0 \):
\[ = \frac{1}{a} \left( -\cos(0) + 0 \cdot \sin(0) \right) = \frac{1}{a} \left( -1 + 0 \right) = -\frac{1}{a} \]

### Final Answer
Thus, the value of the limit is:
\[ \boxed{-\frac{1}{a}} \]