`lim_(xto0) (xcot(4x))/(sin^2x cot^2(2x))` is equal to

Options

A. 2
B. 0
C. 4
D. 1

Answer

To solve the limit \( \lim_{x \to 0} \frac{x \cot(4x)}{\sin^2 x \cot^2(2x)} \), we will follow these steps:

### Step 1: Rewrite cotangent in terms of tangent
Recall that \( \cot(x) = \frac{1}{\tan(x)} \). Therefore, we can rewrite the limit as:
\[ \lim_{x \to 0} \frac{x \cot(4x)}{\sin^2 x \cot^2(2x)} = \lim_{x \to 0} \frac{x \cdot \frac{1}{\tan(4x)}}{\sin^2 x \cdot \left(\frac{1}{\tan(2x)}\right)^2} \]
This simplifies to:
\[ \lim_{x \to 0} \frac{x}{\sin^2 x} \cdot \frac{\tan^2(2x)}{\tan(4x)} \]

### Step 2: Use the limit properties
We know that:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \quad \text{and} \quad \lim_{x \to 0} \frac{\tan x}{x} = 1 \]
Thus, we can express \( \frac{x}{\sin^2 x} \) as:
\[ \frac{x}{\sin^2 x} = \frac{x}{x^2} \cdot \frac{x^2}{\sin^2 x} = \frac{1}{x} \cdot \left(\frac{\sin x}{x}\right)^{-2} \]
As \( x \to 0 \), \( \left(\frac{\sin x}{x}\right)^{-2} \to 1 \), hence:
\[ \lim_{x \to 0} \frac{x}{\sin^2 x} = \infty \]

### Step 3: Analyze the tangent terms
Next, we analyze \( \frac{\tan^2(2x)}{\tan(4x)} \):
\[ \lim_{x \to 0} \frac{\tan(2x)}{2x} = 1 \quad \text{and} \quad \lim_{x \to 0} \frac{\tan(4x)}{4x} = 1 \]
Thus:
\[ \frac{\tan(2x)}{2x} \to 1 \quad \text{and} \quad \frac{\tan(4x)}{4x} \to 1 \]
This means:
\[ \lim_{x \to 0} \frac{\tan^2(2x)}{\tan(4x)} = \lim_{x \to 0} \frac{(2x)^2}{4x} = \lim_{x \to 0} \frac{4x^2}{4x} = \lim_{x \to 0} x = 0 \]

### Step 4: Combine the limits
Now, we combine the limits:
\[ \lim_{x \to 0} \frac{x}{\sin^2 x} \cdot \frac{\tan^2(2x)}{\tan(4x)} = \infty \cdot 0 \]
This form is indeterminate, so we need to evaluate it more carefully.

### Step 5: Final evaluation
We can rewrite the limit as:
\[ \lim_{x \to 0} \frac{x \tan^2(2x)}{\sin^2 x \tan(4x)} \]
Using the previous limits, we can find:
\[ \lim_{x \to 0} \frac{x \cdot 4x^2}{\sin^2 x \cdot 4x} = \lim_{x \to 0} \frac{4x^3}{4x \cdot \sin^2 x} = \lim_{x \to 0} \frac{x^2}{\sin^2 x} \]
As \( x \to 0 \), this limit approaches \( 1 \).

### Final Result
Thus, we conclude:
\[ \lim_{x \to 0} \frac{x \cot(4x)}{\sin^2 x \cot^2(2x)} = 1 \]