`lim_(x to 0) (sin^(2) x//4)/(x)` is equal to

Options

A. 0
B. 1
C. 2
D. `-1`

Answer

To solve the limit \( \lim_{x \to 0} \frac{\sin^2 x / 4}{x} \), we can follow these steps:

### Step 1: Rewrite the limit
We start by rewriting the limit in a more manageable form:
\[ \lim_{x \to 0} \frac{\sin^2 x}{4x} \]

### Step 2: Factor out the constant
Next, we can factor out the constant \( \frac{1}{4} \) from the limit:
\[ \lim_{x \to 0} \frac{\sin^2 x}{4x} = \frac{1}{4} \lim_{x \to 0} \frac{\sin^2 x}{x} \]

### Step 3: Use the limit property
We know from the standard limit property that:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
Thus, we can express \( \sin^2 x \) as:
\[ \sin^2 x = \left(\sin x\right)^2 = \left(\frac{\sin x}{x} \cdot x\right)^2 = \frac{\sin^2 x}{x^2} \cdot x^2 \]
So we can rewrite our limit:
\[ \lim_{x \to 0} \frac{\sin^2 x}{x} = \lim_{x \to 0} \left(\frac{\sin x}{x}\right)^2 \cdot x \]

### Step 4: Evaluate the limit
Now we can evaluate the limit:
\[ \lim_{x \to 0} \left(\frac{\sin x}{x}\right)^2 \cdot x = 1^2 \cdot 0 = 0 \]

### Step 5: Combine results
Putting it all together, we have:
\[ \frac{1}{4} \cdot 0 = 0 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{\sin^2 x / 4}{x} = 0 \]

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