` lim_(x to 0) (x tan 2x -2x tan x)/((1- cos 2x)^(2))` equal

Options

A. 1
B. `1/3`
C. `1/4`
D. `1/2`

Answer

To solve the limit

\[ \lim_{x \to 0} \frac{x \tan 2x - 2x \tan x}{(1 - \cos 2x)^2} \]

we will follow these steps:

### Step 1: Rewrite the limit expression
We start by rewriting the limit expression using the trigonometric identity for \(\tan\) and \(\cos\):

\[ \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \]

Thus, we can express \(x \tan 2x\) as:

\[ x \tan 2x = x \cdot \frac{2 \tan x}{1 - \tan^2 x} \]

### Step 2: Substitute and simplify
Now we substitute this into the limit:

\[ \lim_{x \to 0} \frac{x \cdot \frac{2 \tan x}{1 - \tan^2 x} - 2x \tan x}{(1 - \cos 2x)^2} \]

Factoring out \(2x \tan x\):

\[ = \lim_{x \to 0} \frac{2x \tan x \left( \frac{1}{1 - \tan^2 x} - 1 \right)}{(1 - \cos 2x)^2} \]

### Step 3: Simplify the numerator
Now simplify the term in the parentheses:

\[ \frac{1}{1 - \tan^2 x} - 1 = \frac{1 - (1 - \tan^2 x)}{1 - \tan^2 x} = \frac{\tan^2 x}{1 - \tan^2 x} \]

Thus, the limit becomes:

\[ \lim_{x \to 0} \frac{2x \tan^3 x}{(1 - \cos 2x)^2 (1 - \tan^2 x)} \]

### Step 4: Use the identity for \(\cos\)
Using the identity for \(\cos\):

\[ 1 - \cos 2x = 2 \sin^2 x \]

So we have:

\[ (1 - \cos 2x)^2 = (2 \sin^2 x)^2 = 4 \sin^4 x \]

### Step 5: Substitute back into the limit
Now substituting this back into our limit gives:

\[ = \lim_{x \to 0} \frac{2x \tan^3 x}{4 \sin^4 x (1 - \tan^2 x)} \]

### Step 6: Rewrite \(\tan\) in terms of \(\sin\) and \(\cos\)
We know that:

\[ \tan x = \frac{\sin x}{\cos x} \]

Thus:

\[ \tan^3 x = \frac{\sin^3 x}{\cos^3 x} \]

Substituting this into the limit:

\[ = \lim_{x \to 0} \frac{2x \frac{\sin^3 x}{\cos^3 x}}{4 \sin^4 x (1 - \tan^2 x)} \]

### Step 7: Simplify further
This simplifies to:

\[ = \lim_{x \to 0} \frac{2x \sin^3 x}{4 \sin^4 x \cos^3 x (1 - \tan^2 x)} = \lim_{x \to 0} \frac{2x}{4 \sin x \cos^3 x (1 - \tan^2 x)} \]

### Step 8: Evaluate the limit
As \(x \to 0\), we know:

\[ \frac{\sin x}{x} \to 1 \quad \text{and} \quad \tan^2 x \to 0 \]

Thus, the limit evaluates to:

\[ = \frac{2}{4 \cdot 1 \cdot 1} = \frac{1}{2} \]

### Final Answer
Therefore, the limit is:

\[ \boxed{\frac{1}{2}} \]