Evaluate the following limits :
`Lim_(x to 0) (sin x cos x)/(3x)`

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{\sin x \cos x}{3x} \), we can follow these steps:

### Step 1: Factor out the constant
We start by factoring out the constant \( \frac{1}{3} \) from the limit expression:

\[ \lim_{x \to 0} \frac{\sin x \cos x}{3x} = \frac{1}{3} \lim_{x \to 0} \frac{\sin x \cos x}{x} \]

### Step 2: Rewrite the limit
Next, we can rewrite the limit by separating the sine function:

\[ \frac{1}{3} \lim_{x \to 0} \frac{\sin x \cos x}{x} = \frac{1}{3} \lim_{x \to 0} \left( \frac{\sin x}{x} \cdot \cos x \right) \]

### Step 3: Apply limit properties
Using the property of limits, we can separate the limit into two parts:

\[ \frac{1}{3} \left( \lim_{x \to 0} \frac{\sin x}{x} \cdot \lim_{x \to 0} \cos x \right) \]

### Step 4: Evaluate the limits
We know from standard limit results that:

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
and
\[ \lim_{x \to 0} \cos x = \cos(0) = 1 \]

### Step 5: Substitute the values
Substituting these values back into our expression, we get:

\[ \frac{1}{3} \cdot 1 \cdot 1 = \frac{1}{3} \]

### Final Result
Thus, the limit evaluates to:

\[ \lim_{x \to 0} \frac{\sin x \cos x}{3x} = \frac{1}{3} \]

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