`lim_(x to 0)((1+5x^(2))/(1+3x^(2)))^(1//x^(2))`=

Options

A. e
B. `e^(2)`
C. `e^(3)`
D. none

Answer

To solve the limit \( \lim_{x \to 0} \left( \frac{1 + 5x^2}{1 + 3x^2} \right)^{\frac{1}{x^2}} \), we will follow these steps:

### Step 1: Rewrite the limit
We start by rewriting the limit in a more manageable form:
\[ L = \lim_{x \to 0} \left( \frac{1 + 5x^2}{1 + 3x^2} \right)^{\frac{1}{x^2}} \]

### Step 2: Identify the indeterminate form
As \( x \to 0 \), both the numerator and denominator approach 1, leading to the form \( 1^\infty \), which is an indeterminate form.

### Step 3: Use the exponential limit property
To resolve the indeterminate form, we can use the property:
\[ L = e^{\lim_{x \to 0} \left( \frac{1 + 5x^2}{1 + 3x^2} - 1 \right) \cdot \frac{1}{x^2}} \]

### Step 4: Simplify the expression inside the limit
We need to simplify \( \frac{1 + 5x^2}{1 + 3x^2} - 1 \):
\[ \frac{1 + 5x^2}{1 + 3x^2} - 1 = \frac{1 + 5x^2 - (1 + 3x^2)}{1 + 3x^2} = \frac{5x^2 - 3x^2}{1 + 3x^2} = \frac{2x^2}{1 + 3x^2} \]

### Step 5: Substitute back into the limit
Now, substitute this back into the limit:
\[ L = e^{\lim_{x \to 0} \frac{2x^2}{1 + 3x^2} \cdot \frac{1}{x^2}} = e^{\lim_{x \to 0} \frac{2}{1 + 3x^2}} \]

### Step 6: Evaluate the limit
As \( x \to 0 \), \( 3x^2 \to 0 \), so:
\[ \lim_{x \to 0} \frac{2}{1 + 3x^2} = \frac{2}{1 + 0} = 2 \]

### Step 7: Final result
Thus, we have:
\[ L = e^2 \]

### Conclusion
Therefore, the limit is:
\[ \lim_{x \to 0} \left( \frac{1 + 5x^2}{1 + 3x^2} \right)^{\frac{1}{x^2}} = e^2 \]