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

### Step 2: Multiply and Divide by \( a \)
To facilitate the evaluation, we can multiply and divide the expression by \( a \):
\[ \lim_{x \to 0} \frac{\sin(ax)}{x} = \lim_{x \to 0} \frac{\sin(ax)}{ax} \cdot a \]

### Step 3: Recognize the Standard Limit
Now, we can separate the limit:
\[ = a \cdot \lim_{x \to 0} \frac{\sin(ax)}{ax} \]

### Step 4: Apply the Standard Limit
We know from calculus that:
\[ \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \]
Thus, as \( x \to 0 \), \( ax \to 0 \) as well. Therefore:
\[ \lim_{x \to 0} \frac{\sin(ax)}{ax} = 1 \]

### Step 5: Substitute Back into the Limit
Now substituting back into our limit expression:
\[ = a \cdot 1 = a \]

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

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