Evaluate the following limits :
`Lim_( xto 0 ) (tan 2x)/x `

Answer

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

### Step 1: Rewrite the limit using the double angle formula for tangent
We know that \( \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \). Therefore, we can rewrite the limit as:
\[ \lim_{x \to 0} \frac{\tan 2x}{x} = \lim_{x \to 0} \frac{2 \tan x}{x(1 - \tan^2 x)} \]

### Step 2: Simplify the limit
We can separate the limit into two parts:
\[ \lim_{x \to 0} \frac{2 \tan x}{x} \cdot \frac{1}{1 - \tan^2 x} \]

### Step 3: Evaluate the first limit
Using the known limit \( \lim_{x \to 0} \frac{\tan x}{x} = 1 \), we have:
\[ \lim_{x \to 0} \frac{2 \tan x}{x} = 2 \cdot 1 = 2 \]

### Step 4: Evaluate the second limit
Next, we need to evaluate \( \lim_{x \to 0} \frac{1}{1 - \tan^2 x} \). As \( x \to 0 \), \( \tan x \to 0 \), hence \( \tan^2 x \to 0 \):
\[ \lim_{x \to 0} \frac{1}{1 - \tan^2 x} = \frac{1}{1 - 0} = 1 \]

### Step 5: Combine the results
Now we combine the results from Step 3 and Step 4:
\[ \lim_{x \to 0} \frac{\tan 2x}{x} = 2 \cdot 1 = 2 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{\tan 2x}{x} = 2 \]

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