The value of the limit
`lim_(x to 0) ((x)/(sinx))^(6//x^(2))` is

Options

A. e
B. `e^(-1)`
C. `e^(-1//6)`
D. `e^(6)`

Explanation

To solve the limit

\[ \lim_{x \to 0} \left( \frac{x}{\sin x} \right)^{\frac{6}{x^2}}, \]

we can follow these steps:

### Step 1: Identify the form of the limit
As \( x \to 0 \), we know that \( \sin x \) approaches \( x \). Thus, \( \frac{x}{\sin x} \) approaches \( 1 \). However, since we are raising this expression to the power of \( \frac{6}{x^2} \), we are dealing with an indeterminate form of type \( 1^\infty \).

**Hint:** Recognize the indeterminate form and consider taking the natural logarithm to simplify the limit.

### Step 2: Take the natural logarithm
Let

\[ L = \lim_{x \to 0} \left( \frac{x}{\sin x} \right)^{\frac{6}{x^2}}. \]

Taking the natural logarithm, we have:

\[ \ln L = \lim_{x \to 0} \frac{6}{x^2} \ln \left( \frac{x}{\sin x} \right). \]

**Hint:** Rewrite \( \ln \left( \frac{x}{\sin x} \right) \) using properties of logarithms.

### Step 3: Simplify the logarithm
We can rewrite:

\[ \ln \left( \frac{x}{\sin x} \right) = \ln x - \ln(\sin x). \]

Thus, we have:

\[ \ln L = \lim_{x \to 0} \frac{6}{x^2} \left( \ln x - \ln(\sin x) \right). \]

**Hint:** Use the Taylor series expansion for \( \sin x \) around \( x = 0 \).

### Step 4: Use Taylor series for \( \sin x \)
The Taylor series expansion of \( \sin x \) is:

\[ \sin x = x - \frac{x^3}{6} + O(x^5). \]

Thus,

\[ \ln(\sin x) \approx \ln\left(x - \frac{x^3}{6}\right) \approx \ln x + \ln\left(1 - \frac{x^2}{6}\right) \approx \ln x - \frac{x^2}{6} + O(x^4). \]

**Hint:** Substitute this approximation back into the limit.

### Step 5: Substitute and simplify
Now substituting back, we get:

\[ \ln L = \lim_{x \to 0} \frac{6}{x^2} \left( \ln x - \left( \ln x - \frac{x^2}{6} \right) \right) = \lim_{x \to 0} \frac{6}{x^2} \left( \frac{x^2}{6} \right) = \lim_{x \to 0} 1 = 1. \]

**Hint:** Exponentiate to find \( L \).

### Step 6: Exponentiate to find \( L \)
Since \( \ln L = 1 \), we have:

\[ L = e^1 = e. \]

Thus, the value of the limit is

\[ \boxed{e}. \]