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The given expression is:

\[
\lim_{{x \to 0}} \left( 6x^2 \cot(x) \csc(2x) \right)
\]

To solve this limit, let's break it down and analyze it step by step.

### Step 1: Rewrite cot(x) and csc(2x)
We know the following trigonometric identities:
- \(\cot(x) = \frac{\cos(x)}{\sin(x)}\)
- \(\csc(2x) = \frac{1}{\sin(2x)}\)
  Using the double-angle identity for sine, \(\sin(2x) = 2\sin(x)\cos(x)\).

Thus, the original expression becomes:
\[
6x^2 \cdot \frac{\cos(x)}{\sin(x)} \cdot \frac{1}{\sin(2x)} = 6x^2 \cdot \frac{\cos(x)}{\sin(x)} \cdot \frac{1}{2\sin(x)\cos(x)}
\]

Simplifying further:
\[
= \frac{6x^2}{2\sin^2(x)} = 3x^2 \cdot \frac{1}{\sin^2(x)}
\]

### Step 2: Apply the limit
We need to evaluate the following limit:
\[
\lim_{{x \to 0}} 3x^2 \cdot \frac{1}{\sin^2(x)}
\]

Using the standard limit identity:
\[
\lim_{{x \to 0}} \frac{x}{\sin(x)} = 1
\]
We get:
\[
\lim_{{x \to 0}} 3 \cdot \left( \frac{x}{\sin(x)} \right)^2 = 3 \cdot 1^2 = 3
\]

Thus, the value of the limit is \(3\).

### Final Answer:
\[
\lim_{{x \to 0}} \left( 6x^2 \cot(x) \csc(2x) \right) = 3
\]

Do you want more details or have any questions?

Here are 5 related questions for practice:
1. What is the limit \(\lim_{{x \to 0}} \frac{\sin(x)}{x}\)?
2. How do you simplify \(\csc(2x)\) using trigonometric identities?
3. What is the limit \(\lim_{{x \to 0}} \frac{x^2}{\sin(x)}\)?
4. What are the standard trigonometric limits as \(x\) approaches zero?
5. How do you use L'Hopital's Rule to evaluate indeterminate forms?

**Tip:** When dealing with trigonometric limits, applying known limit identities like \(\lim_{{x \to 0}} \frac{\sin(x)}{x} = 1\) can simplify the solution significantly.

Calculate the limit as x approaches 0 for the expression 6x^2 cot(x) csc(2x).

Explore the step-by-step solution to calculate the limit of the expression 6x^2 cot(x) csc(2x) as x approaches 0. This guide covers the application of trigonometric identities and limits, particularly focusing on the behavior of trigonometric functions near zero.

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