Evaluate the following limits:
`lim_(xrarr0)(tan(x//2))/(3x)`

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{\tan\left(\frac{x}{2}\right)}{3x} \), we can follow these steps:

### Step 1: Rewrite the limit
We start with the limit:
\[ \lim_{x \to 0} \frac{\tan\left(\frac{x}{2}\right)}{3x} \]

### Step 2: Use the limit property of \(\tan\)
As \( x \to 0 \), we can use the property that \( \tan(kx) \approx kx \) for small values of \( x \). Specifically, we have:
\[ \tan\left(\frac{x}{2}\right) \approx \frac{x}{2} \quad \text{as } x \to 0 \]

### Step 3: Substitute the approximation into the limit
Substituting this approximation into our limit gives:
\[ \lim_{x \to 0} \frac{\frac{x}{2}}{3x} \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ \frac{\frac{x}{2}}{3x} = \frac{x}{2} \cdot \frac{1}{3x} = \frac{1}{6} \]

### Step 5: Evaluate the limit
Since there are no \( x \) terms left in the expression, we can directly evaluate the limit:
\[ \lim_{x \to 0} \frac{1}{6} = \frac{1}{6} \]

### Final Answer
Thus, the limit is:
\[ \frac{1}{6} \]

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