`lim_(x to 0)((tan x)/x)^(1//x^(2))`=

Options

A. `a^(2)`
B. e
C. `sqrt(e)`
D. `e^(1//3)`

Answer

To solve the limit \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} \), we can follow these steps:

### Step 1: Identify the limit form
First, we recognize that as \( x \to 0 \), \( \tan x \) approaches \( x \). Therefore, we can rewrite the limit:
\[ \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} \to \left( \frac{0}{0} \right)^{\infty} \]
This is an indeterminate form of type \( 1^\infty \).

### Step 2: Rewrite using the exponential function
We can use the property of limits that allows us to express this limit in terms of the exponential function:
\[ \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = e^{\lim_{x \to 0} \frac{1}{x^2} \left( \frac{\tan x}{x} - 1 \right)} \]

### Step 3: Simplify the expression inside the limit
Next, we need to evaluate the limit:
\[ \lim_{x \to 0} \frac{1}{x^2} \left( \frac{\tan x}{x} - 1 \right) \]
Using the Taylor series expansion for \( \tan x \), we have:
\[ \tan x = x + \frac{x^3}{3} + O(x^5) \]
Thus,
\[ \frac{\tan x}{x} = 1 + \frac{x^2}{3} + O(x^4) \]
This implies:
\[ \frac{\tan x}{x} - 1 = \frac{x^2}{3} + O(x^4) \]

### Step 4: Substitute back into the limit
Now substituting back into our limit:
\[ \lim_{x \to 0} \frac{1}{x^2} \left( \frac{\tan x}{x} - 1 \right) = \lim_{x \to 0} \frac{1}{x^2} \left( \frac{x^2}{3} + O(x^4) \right) = \lim_{x \to 0} \left( \frac{1}{3} + O(x^2) \right) = \frac{1}{3} \]

### Step 5: Final limit calculation
Now we can substitute this result back into the exponential:
\[ \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = e^{\frac{1}{3}} \]

### Conclusion
Thus, the final answer is:
\[ \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = e^{\frac{1}{3}} \]