Question:

<p><strong>Step 1: Recognize the standard limit form.</strong></p><p>We have a limit of the form $\lim_{x \to 0}\left(1 + \frac{f(x) + x^2}{x^2}\right)^{1/x}$.</p><p>For limits of type $\lim (1 + u)^{1/v} = e^k$, we use: if $u \to 0$ and $u/v \to k$, then $(1+u)^{1/v} \to e^k$.</p><p><strong>Step 2: Set up the condition.</strong></p><p>Let $u = \frac{f(x) + x^2}{x^2}$. For the limit to equal $e^2$, we need:</p><p>$$\lim_{x \to 0} \frac{u}{1/x} = 2$$</p><p>This means: $\lim_{x \to 0} u \cdot x = 2$</p><p>$$\lim_{x \to 0} \frac{f(x) + x^2}{x^2} \cdot x = 2$$</p><p>$$\lim_{x \to 0} \frac{f(x) + x^2}{x} = 2$$</p><p><strong>Step 3: Analyze the polynomial f(x).</strong></p><p>For this limit to exist and equal 2, the numerator $f(x) + x^2$ must be divisible by $x$ (otherwise the limit is infinite or undefined).</p><p>For least degree polynomial: let $f(x) = ax + bx^2 + cx^3 + \ldots$</p><p>Then: $f(x) + x^2 = ax + (b+1)x^2 + cx^3 + \ldots$</p><p><strong>Step 4: Apply the limit condition.</strong></p><p>$$\lim_{x \to 0} \frac{ax + (b+1)x^2 + cx^3 + \ldots}{x} = \lim_{x \to 0} \left(a + (b+1)x + cx^2 + \ldots\right) = a = 2$$</p><p>So $a = 2$, and $f(x)$ must have no constant term for least degree.</p><p><strong>Step 5: Determine the minimal form.</strong></p><p>For the polynomial to have least degree with the leading coefficient being 2: $f(x) = 2x$</p><p>We can verify: $\lim_{x \to 0} \frac{2x + x^2}{x} = \lim_{x \to 0}(2 + x) = 2$ ✓</p><p><strong>Step 6: Calculate f(2).</strong></p><p>$$f(2) = 2(2) = 4$$</p><p>∴ Answer: 4</p>