What is the value of `lim_(xto0)(sinx^(0))/(tan3x^(0))` ?

Options

A. `(1)/(4)`
B. `(1)/(3)`
C. `(1)/(2)`
D. 1

Answer

To solve the limit \( \lim_{x \to 0} \frac{\sin(x^0)}{\tan(3x^0)} \), we first need to clarify the expression. The notation \( x^0 \) is equal to 1 for any \( x \neq 0 \). Therefore, we can rewrite the limit as:

\[ \lim_{x \to 0} \frac{\sin(1)}{\tan(3)} \]

Since both \( \sin(1) \) and \( \tan(3) \) are constants, we can directly evaluate the limit:

1. **Evaluate \( \sin(1) \)**: The sine of 1 (in radians) is a constant value.
2. **Evaluate \( \tan(3) \)**: The tangent of 3 (in radians) is also a constant value.

Thus, the limit simplifies to:

\[ \frac{\sin(1)}{\tan(3)} \]

Since both sine and tangent functions are continuous, we can directly substitute the values:

\[ \lim_{x \to 0} \frac{\sin(1)}{\tan(3)} = \frac{\sin(1)}{\tan(3)} \]

However, if we were to consider the limit as \( x \to 0 \) in a different context (for example, if we were to consider \( \sin(x) \) and \( \tan(3x) \)), we would use the small angle approximations:

1. **Using the small angle approximation**: As \( x \to 0 \), \( \sin(x) \approx x \) and \( \tan(3x) \approx 3x \).
2. **Rewrite the limit**: The limit would then be rewritten as:

\[ \lim_{x \to 0} \frac{x}{3x} = \frac{1}{3} \]

Thus, the final value of the limit is:

\[ \frac{1}{3} \]

### Summary of Steps:
1. Recognize that \( x^0 = 1 \) for \( x \neq 0 \).
2. Rewrite the limit as \( \frac{\sin(1)}{\tan(3)} \).
3. If considering small angles, use approximations \( \sin(x) \approx x \) and \( \tan(3x) \approx 3x \).
4. Simplify the limit to find \( \frac{1}{3} \).