Evaluate `lim_(x to 0) ("sin"ax)/(x)`

Answer

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

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

### Step 2: Check the form of the limit
As \( x \to 0 \), both the numerator \( \sin(ax) \) and the denominator \( x \) approach 0. This gives us the indeterminate form \( \frac{0}{0} \), which allows us to apply L'Hôpital's Rule.

### Step 3: Apply L'Hôpital's Rule
According to L'Hôpital's Rule, we differentiate the numerator and the denominator:
- The derivative of the numerator \( \sin(ax) \) is \( a \cos(ax) \) (using the chain rule).
- The derivative of the denominator \( x \) is \( 1 \).

Now we rewrite the limit:
\[ \lim_{x \to 0} \frac{\sin(ax)}{x} = \lim_{x \to 0} \frac{a \cos(ax)}{1} \]

### Step 4: Evaluate the limit
Now we can directly substitute \( x = 0 \):
\[ = a \cos(a \cdot 0) = a \cos(0) = a \cdot 1 = a \]

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

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