Deduction : `lim_(x to 0)(sin x^(0))/x` =

Options

A. 1
B. `pi`
C. `pi//180`
D. none

Answer

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

### Step 1: Convert Degrees to Radians
First, we need to convert \( x^\circ \) (degrees) into radians. The conversion formula is:
\[ x^\circ = x \cdot \frac{\pi}{180} \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 0} \frac{\sin\left(x \cdot \frac{\pi}{180}\right)}{x} \]

### Step 2: Substitute the Radian Conversion
Now, substituting the conversion into the limit gives us:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{\pi x}{180}\right)}{x} \]

### Step 3: Factor Out the Constant
We can factor out the constant \( \frac{\pi}{180} \) from the sine function:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{\pi x}{180}\right)}{x} = \lim_{x \to 0} \frac{\sin\left(\frac{\pi x}{180}\right)}{\frac{\pi x}{180}} \cdot \frac{\pi}{180} \]

### Step 4: Apply the Standard Limit
Using the standard limit \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \), we can set \( u = \frac{\pi x}{180} \). As \( x \to 0 \), \( u \to 0 \) as well. Therefore, we have:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{\pi x}{180}\right)}{\frac{\pi x}{180}} = 1 \]

### Step 5: Combine the Results
Now we can combine our results:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{\pi x}{180}\right)}{x} = 1 \cdot \frac{\pi}{180} = \frac{\pi}{180} \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{\sin(x^\circ)}{x} = \frac{\pi}{180} \]

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