Evaluate: `lim_(xto0)(tan8x)/(sin3x)`

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{\tan(8x)}{\sin(3x)} \), we can follow these steps:

### Step 1: Rewrite the limit
We start by rewriting the limit expression:
\[ \lim_{x \to 0} \frac{\tan(8x)}{\sin(3x)} = \lim_{x \to 0} \frac{\tan(8x)}{8x} \cdot \frac{8x}{\sin(3x)} \]
This allows us to separate the limit into two parts.

### Step 2: Apply standard limits
Using the standard limits:
\[ \lim_{x \to 0} \frac{\tan(ax)}{ax} = 1 \quad \text{and} \quad \lim_{x \to 0} \frac{\sin(ax)}{ax} = 1 \]
we can evaluate each part:
\[ \lim_{x \to 0} \frac{\tan(8x)}{8x} = 1 \quad \text{and} \quad \lim_{x \to 0} \frac{\sin(3x)}{3x} = 1 \]

### Step 3: Rewrite the limit with constants
To utilize these limits, we rewrite our expression:
\[ \lim_{x \to 0} \frac{\tan(8x)}{\sin(3x)} = \lim_{x \to 0} \left( \frac{\tan(8x)}{8x} \cdot \frac{8}{\frac{\sin(3x)}{3x}} \cdot \frac{3x}{3} \right) \]
This gives us:
\[ = \lim_{x \to 0} \frac{\tan(8x)}{8x} \cdot \frac{8}{\frac{\sin(3x)}{3x}} \cdot 3 \]

### Step 4: Evaluate the limits
Now we can evaluate the limits:
\[ = 1 \cdot \frac{8}{1} \cdot 3 = 8 \cdot 3 = 24 \]

### Final Answer
Thus, the limit evaluates to:
\[ \lim_{x \to 0} \frac{\tan(8x)}{\sin(3x)} = \frac{8}{3} \]