`lim_(xrarr0) (cos ec x -cot x)`

Answer

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

### Step 1: Rewrite the expression
We start with the limit:
\[ \lim_{x \to 0} \left( \cos x - \cot x \right) \]
Recall that \( \cot x = \frac{\cos x}{\sin x} \). Therefore, we can rewrite the expression as:
\[ \lim_{x \to 0} \left( \cos x - \frac{\cos x}{\sin x} \right) \]

### Step 2: Combine the terms
To combine the terms, we need a common denominator. The common denominator here is \( \sin x \):
\[ \lim_{x \to 0} \left( \frac{\cos x \sin x - \cos x}{\sin x} \right) \]
This simplifies to:
\[ \lim_{x \to 0} \frac{\cos x (\sin x - 1)}{\sin x} \]

### Step 3: Evaluate the limit
Now, we substitute \( x = 0 \):
\[ \frac{\cos(0)(\sin(0) - 1)}{\sin(0)} = \frac{1(0 - 1)}{0} = \frac{-1}{0} \]
This is an indeterminate form \( \frac{0}{0} \).

### Step 4: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that we can differentiate the numerator and the denominator:
\[ \lim_{x \to 0} \frac{\frac{d}{dx}[\cos x (\sin x - 1)]}{\frac{d}{dx}[\sin x]} \]

### Step 5: Differentiate the numerator and denominator
Differentiating the numerator using the product rule:
\[ \frac{d}{dx}[\cos x (\sin x - 1)] = -\sin x (\sin x - 1) + \cos x \cos x = -\sin^2 x + \cos^2 x \]
The derivative of the denominator is:
\[ \frac{d}{dx}[\sin x] = \cos x \]

### Step 6: Substitute back into the limit
Now we have:
\[ \lim_{x \to 0} \frac{-\sin^2 x + \cos^2 x}{\cos x} \]
Substituting \( x = 0 \):
\[ \frac{-\sin^2(0) + \cos^2(0)}{\cos(0)} = \frac{0 + 1}{1} = 1 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \left( \cos x - \cot x \right) = 1 \]