Let f:[0,∞)→[0,∞) be a function satisfying f(x)=xe−f(x)∀x∈[0,∞), then which of the following is NOT CORRECT?

lim x → ∞ f ( x ) ln x = 1 e

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Let $f : [0, \infty) \to [0, \infty)$ be a function satisfying $f (x) = x e^{- f (x)} \forall x \in [0, \infty),$ then which of the following is NOT CORRECT?

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$\lim_{x \to \infty} \frac{f (x)}{\ln x} = \frac{1}{e}$

$\lim_{x \to \infty} f (x) = \infty$

$f (x)$ is onto function.

$f (x)$ is strictly increasing.

answer is C.

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Let y = f(x), then $x = y e^{y}$

$\lim_{x \to \infty} \frac{f (x)}{l n x} = \lim_{y \to \infty} \frac{y}{l n (y e^{y})} = \lim_{y \to \infty} \frac{y}{l n y + y} = 1$

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