`lim_(xrarr0) (sinax)/(bx)`

Explanation

To solve the limit \( \lim_{x \to 0} \frac{\sin(ax)}{bx} \), we can follow these steps:

### Step 1: Rewrite the Limit
We start with the limit expression:
\[ \lim_{x \to 0} \frac{\sin(ax)}{bx} \]

### Step 2: Factor out Constants
We can factor out the constant \( b \) from the denominator:
\[ = \lim_{x \to 0} \frac{1}{b} \cdot \frac{\sin(ax)}{x} \]

### Step 3: Multiply and Divide by \( a \)
Next, we multiply and divide the numerator by \( a \):
\[ = \frac{1}{b} \cdot \lim_{x \to 0} \frac{\sin(ax)}{ax} \cdot a \]

### Step 4: Apply the Limit Property
We know from the limit property that:
\[ \lim_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1 \]
Thus, we can substitute \( \theta = ax \):
\[ = \frac{1}{b} \cdot a \cdot \lim_{x \to 0} \frac{\sin(ax)}{ax} = \frac{1}{b} \cdot a \cdot 1 \]

### Step 5: Simplify the Expression
Now we simplify the expression:
\[ = \frac{a}{b} \]

### Final Answer
Therefore, the limit is:
\[ \lim_{x \to 0} \frac{\sin(ax)}{bx} = \frac{a}{b} \]

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