Evaluate the following limits:
`lim_(xrarr0)(x cot2x)`

Explanation

To evaluate the limit \( \lim_{x \to 0} (x \cot(2x)) \), we can follow these steps:

### Step 1: Rewrite cotangent in terms of tangent
We know that:
\[ \cot(2x) = \frac{1}{\tan(2x)} \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 0} (x \cot(2x)) = \lim_{x \to 0} \left( \frac{x}{\tan(2x)} \right) \]

### Step 2: Apply the limit property
We use the limit property:
\[ \lim_{u \to 0} \frac{\tan(u)}{u} = 1 \]
In our case, we need to express \( \tan(2x) \) in a suitable form:
\[ \lim_{x \to 0} \frac{x}{\tan(2x)} = \lim_{x \to 0} \frac{x}{2x} \cdot \frac{2x}{\tan(2x)} \]
This simplifies to:
\[ \lim_{x \to 0} \frac{1}{2} \cdot \frac{2x}{\tan(2x)} \]

### Step 3: Evaluate the limit
Now, we can apply the limit property:
\[ \lim_{x \to 0} \frac{2x}{\tan(2x)} = 1 \]
Thus, we have:
\[ \lim_{x \to 0} (x \cot(2x)) = \frac{1}{2} \cdot 1 = \frac{1}{2} \]

### Conclusion
Therefore, the limit is:
\[ \lim_{x \to 0} (x \cot(2x)) = \frac{1}{2} \]

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