What is `lim_(xtoo)(sin^(2)ax)/(bx)` (a.b are cpnstants) equal to ?

Options

A. 0
B. a
C. `(a^(2))/(b)`
D. Does not exist

Answer

To solve the limit \( \lim_{x \to 0} \frac{\sin^2(ax)}{bx} \), we can follow these steps:

### Step 1: Rewrite the limit
We start with the limit expression:
\[ \lim_{x \to 0} \frac{\sin^2(ax)}{bx} \]
We can rewrite this as:
\[ \lim_{x \to 0} \frac{\sin(ax)}{ax} \cdot \frac{\sin(ax)}{ax} \cdot \frac{a^2x^2}{bx} \]

### Step 2: Factor out constants
Now we can factor out the constants:
\[ = \lim_{x \to 0} \left( \frac{\sin(ax)}{ax} \right)^2 \cdot \frac{a^2x^2}{bx} \]

### Step 3: Simplify the limit
Now we can simplify the limit:
\[ = \lim_{x \to 0} \left( \frac{\sin(ax)}{ax} \right)^2 \cdot \frac{a^2x}{b} \]
As \( x \to 0 \), we know that \( \frac{\sin(ax)}{ax} \to 1 \). Therefore:
\[ = 1^2 \cdot \lim_{x \to 0} \frac{a^2x}{b} = \lim_{x \to 0} \frac{a^2x}{b} \]

### Step 4: Evaluate the limit
Now we evaluate the limit:
\[ = \frac{a^2 \cdot 0}{b} = 0 \]

### Conclusion
Thus, the limit is:
\[ \lim_{x \to 0} \frac{\sin^2(ax)}{bx} = 0 \]