`lim_(xto0) ((1-cos2x)(3+cosx))/(xtan4x)` is equal to

Explanation

To solve the limit \( \lim_{x \to 0} \frac{(1 - \cos(2x))(3 + \cos x)}{x \tan(4x)} \), we can follow these steps:

### Step 1: Identify the form of the limit
First, we substitute \( x = 0 \) into the expression:
- The numerator becomes \( 1 - \cos(0) = 1 - 1 = 0 \).
- The denominator becomes \( 0 \cdot \tan(0) = 0 \).

This gives us a \( \frac{0}{0} \) indeterminate form, so we need to simplify the expression.

**Hint:** Always check the limit by substituting the value first to determine if it's an indeterminate form.

### Step 2: Simplify the numerator
Using the trigonometric identity, we know:
\[ 1 - \cos(2x) = 2 \sin^2(x) \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 0} \frac{(2 \sin^2(x))(3 + \cos x)}{x \tan(4x)} \]

**Hint:** Use trigonometric identities to simplify expressions involving cosine.

### Step 3: Rewrite the tangent function
Recall that \( \tan(4x) = \frac{\sin(4x)}{\cos(4x)} \). Therefore, we can express the limit as:
\[ \lim_{x \to 0} \frac{2 \sin^2(x)(3 + \cos x)}{x \cdot \frac{\sin(4x)}{\cos(4x)}} \]
This simplifies to:
\[ \lim_{x \to 0} \frac{2 \sin^2(x)(3 + \cos x) \cos(4x)}{x \sin(4x)} \]

**Hint:** Remember to express tangent in terms of sine and cosine for simplification.

### Step 4: Factor out the limit
Now, we can separate the limit:
\[ \lim_{x \to 0} \frac{2 \sin^2(x)}{x} \cdot \frac{(3 + \cos x) \cos(4x)}{\sin(4x)} \]

### Step 5: Apply limit properties
Using the known limits:
- \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \)
- \( \lim_{x \to 0} \frac{\sin(4x)}{4x} = 1 \)

We can rewrite \( \frac{2 \sin^2(x)}{x} \) as:
\[ 2 \cdot \frac{\sin^2(x)}{x^2} \cdot x \]
Thus, the limit becomes:
\[ \lim_{x \to 0} 2 \cdot \frac{\sin^2(x)}{x^2} \cdot (3 + \cos x) \cdot \frac{\cos(4x)}{4} \cdot \frac{4}{\sin(4x)} \]

### Step 6: Evaluate the limit
As \( x \to 0 \):
- \( \frac{\sin^2(x)}{x^2} \to 1 \)
- \( 3 + \cos(0) = 3 + 1 = 4 \)
- \( \cos(0) = 1 \)
- \( \frac{4}{\sin(4x)} \to 1 \)

Putting it all together:
\[ L = 2 \cdot 1 \cdot 4 \cdot 1 \cdot 1 = 8 \]

### Final Result
Thus, the limit is:
\[ \boxed{2} \]