Let f : [ 0 , ∞ ) → [ 0 , ∞ ) be bijective and satisfy f ( x ) e f ( x ) = x . Find lim x → ∞ l n x f ( x ) .

Write y = f ( x ) . Then y e y = x , so ln y + y = ln x . Since y → ∞ , division by y gives ln x / y = 1 + ( ln y ) / y → 1 . Hence f ( x ) / ln x → 1 .

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Mathematics - Limits Question with Solution | TestHub

MathematicsLimitsExponential and logarithmic limitsMedium2 minQB

MathematicsMediumsingle choice

Let $f : [0, \infty) \to [0, \infty)$ be bijective and satisfy $f (x) e^{f (x)} = x$ . Find $lim_{x \to \infty} \frac{f ( x )}{l n x}$ .

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Solution:

Write $y = f (x)$ . Then $y e^{y} = x$ , so $ln y + y = ln x$ . Since $y \to \infty$ , division by $y$ gives $ln x / y = 1 + (ln y) / y \to 1$ . Hence $f (x) / ln x \to 1$ .

Stream:JEESubject:MathematicsTopic:LimitsSubtopic:Exponential and logarithmic limits

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