The polynomial of least degree such that `lim_(xto0)(1+(x^(2)+f(x))/(x^(2)))^(1//x)=e^(2)` is

Options

A. `x^(2)`
B. `x^(3)+2x^(2)`
C. `-x^(2)+2x^(3)`
D. None of these

Answer

To solve the problem, we need to find the polynomial \( f(x) \) of the least degree such that

\[ \lim_{x \to 0} \left(1 + \frac{x^2 + f(x)}{x^2}\right)^{\frac{1}{x}} = e^2. \]

### Step 1: Rewrite the limit expression

We start by rewriting the limit expression:

\[ \lim_{x \to 0} \left(1 + \frac{x^2 + f(x)}{x^2}\right)^{\frac{1}{x}} = \lim_{x \to 0} \left(1 + 1 + \frac{f(x)}{x^2}\right)^{\frac{1}{x}} = \lim_{x \to 0} \left(2 + \frac{f(x)}{x^2}\right)^{\frac{1}{x}}. \]

### Step 2: Set up the limit condition

For the limit to equal \( e^2 \), we need:

\[ \lim_{x \to 0} \left(2 + \frac{f(x)}{x^2}\right)^{\frac{1}{x}} = e^2. \]

This implies:

\[ \lim_{x \to 0} \frac{f(x)}{x^2} \to 0 \quad \text{and} \quad \lim_{x \to 0} \left(2 + \frac{f(x)}{x^2}\right)^{\frac{1}{x}} \to e^2. \]

### Step 3: Use the exponential limit property

Using the property of limits, we can express the limit in terms of the exponent:

\[ \lim_{x \to 0} \frac{f(x)}{x^2} = 0 \implies f(x) = o(x^2). \]

Thus, we can write:

\[ \lim_{x \to 0} \frac{f(x)}{x^2} = 2 - 1 = 1. \]

### Step 4: Assume a polynomial form for \( f(x) \)

Let \( f(x) \) be a polynomial of the form:

\[ f(x) = a_1 x^2 + a_2 x^3 + \ldots \]

### Step 5: Evaluate the limit

Substituting \( f(x) \) into the limit gives:

\[ \lim_{x \to 0} \left(2 + \frac{a_1 x^2 + a_2 x^3 + \ldots}{x^2}\right)^{\frac{1}{x}} = \lim_{x \to 0} \left(2 + a_1 + a_2 x + \ldots\right)^{\frac{1}{x}}. \]

### Step 6: Set the coefficients

For the limit to equal \( e^2 \), we need:

\[ 2 + a_1 = 0 \quad \text{and} \quad a_2 = 2. \]

### Step 7: Solve for coefficients

From \( 2 + a_1 = 0 \), we find:

\[ a_1 = -2. \]

And from \( a_2 = 2 \), we have:

\[ a_2 = 2. \]

### Step 8: Write the polynomial

Thus, the polynomial \( f(x) \) of least degree is:

\[ f(x) = -2x^2 + 2x^3. \]

### Final Answer

The polynomial of least degree such that

\[ \lim_{x \to 0} \left(1 + \frac{x^2 + f(x)}{x^2}\right)^{\frac{1}{x}} = e^2 \]

is

\[ f(x) = -x^2 + 2x^3. \]