Evaluate the following limits :
`Lim_(x to 0) (tan. 1/2x)/(3x)`

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{\tan\left(\frac{1}{2}x\right)}{3x} \), we can follow these steps:

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

### Step 2: Multiply and Divide by \(\frac{1}{2}\)
To make use of the known limit \(\lim_{u \to 0} \frac{\tan(u)}{u} = 1\), we can multiply and divide the expression by \(\frac{1}{2}\):
\[ \lim_{x \to 0} \frac{\tan\left(\frac{1}{2}x\right)}{3x} = \lim_{x \to 0} \frac{\tan\left(\frac{1}{2}x\right)}{\frac{1}{2}x} \cdot \frac{1}{3 \cdot \frac{1}{2}} = \lim_{x \to 0} \frac{\tan\left(\frac{1}{2}x\right)}{\frac{1}{2}x} \cdot \frac{2}{3} \]

### Step 3: Apply the Limit Identity
Using the limit identity \(\lim_{u \to 0} \frac{\tan(u)}{u} = 1\), we can evaluate the limit:
\[ \lim_{x \to 0} \frac{\tan\left(\frac{1}{2}x\right)}{\frac{1}{2}x} = 1 \]

### Step 4: Substitute Back into the Expression
Now substituting back into our expression, we have:
\[ \frac{2}{3} \cdot 1 = \frac{2}{3} \]

### Final Answer
Thus, the limit evaluates to:
\[ \lim_{x \to 0} \frac{\tan\left(\frac{1}{2}x\right)}{3x} = \frac{2}{3} \]