Evaluate the following limits :
`Lim_(x to 0 ) ((sin ax)/(sin bx))^(k)`

Answer

To evaluate the limit \( L = \lim_{x \to 0} \left( \frac{\sin(ax)}{\sin(bx)} \right)^k \), we can follow these steps:

### Step 1: Rewrite the Limit
We start by rewriting the limit:
\[ L = \lim_{x \to 0} \left( \frac{\sin(ax)}{\sin(bx)} \right)^k \]
This can be expressed as:
\[ L = \lim_{x \to 0} \left( \frac{\sin(ax)}{ax} \cdot \frac{ax}{bx} \cdot \frac{bx}{\sin(bx)} \right)^k \]

### Step 2: Apply the Limit
As \( x \to 0 \), both \( \frac{\sin(ax)}{ax} \) and \( \frac{\sin(bx)}{bx} \) approach 1. Therefore, we can write:
\[ L = \lim_{x \to 0} \left( \frac{\sin(ax)}{ax} \cdot \frac{a}{b} \cdot \frac{bx}{\sin(bx)} \right)^k \]
This simplifies to:
\[ L = \left( \frac{a}{b} \cdot 1 \cdot 1 \right)^k \]

### Step 3: Final Result
Thus, we obtain:
\[ L = \left( \frac{a}{b} \right)^k \]

### Conclusion
The final answer is:
\[ L = \left( \frac{a}{b} \right)^k \]

---