Limits, Continuity & Differentiability Question - None

Question

If $$f(x)$$ is a polynomial of least degree such that $\lim_{x \to 0}\left(1 + \dfrac{f(x) + x^2}{x^2}\right)^{1/x} = e^2$ , then $$f(2)$$ is

Accepted Answer

Key Idea: For $\lim_{x \to 0}\left(1 + \dfrac{f(x) + x^2}{x^2}\right)^{1/x} = e^2$ form: $$f(2)$$ , so $\lim_{x o 0}\dfrac{f(x)+x^2}{x^3} = 2$ .

Step 1: For the limit to be finite, the numerator must vanish faster than $$x^3$$ . So $$f(x) + x^2$$ must be $$O(x^3)$$ , meaning the $$x^2$$ term in $$f(x)$$ must cancel the $$+x^2$$ .

Step 2: Least degree polynomial means no unnecessary terms. Try $$f(x) = 2x^3 - x^2$$ :
$\lim_{x o 0}\frac{2x^3 - x^2 + x^2}{x^3} = \lim_{x o 0}\frac{2x^3}{x^3} = 2 \checkmark$

Step 3: Compute $$f(2)$$ :
$$$f(2) = 2(8) - 4 = 16 - 4 = 12$$$

Answer: (D) 12

Trap: Students try

$f(x) = 2x^3$

(forgetting to cancel the

$x^2$

term), which gives

\lim \frac{2x^3+x^2}{x^3} = 2 + \frac{1}{x} o \infty

. The

$-x^2$

is essential.

Key Concept: Least degree polynomial construction using limit finiteness condition

Question:

Step-by-Step Solution

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