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EEJ MAIN Mathematics QUESTION #6012
Question 1

Let $f$ be differentiable with $f'(x)=7-\dfrac{3}{4}\dfrac{f(x)}{x}$ for $x>0$ and $f(1)\neq 4$. Find $\displaystyle\lim_{x\to 0^+} x\,f\!\left(\dfrac{1}{x}\right)$.

  • Exists and equals $\dfrac{4}{7}$
  • Exists and equals $4$✔️
  • Does not exist
  • Exists and equals $0$
Correct Answer Explanation

Let $g(x) = f(x)/x$. Substituting $t = 1/x$ (so $x\to 0^+$ means $t\to\infty$):

$x\cdot f(1/x) = \frac{f(t)}{t} = g(t)$ as $t\to\infty$.

From $f'(x) = 7 - \frac{3}{4}\frac{f(x)}{x}$, the equilibrium $g = f(x)/x$ satisfies: as $x\to\infty$, $g\to 4$ (the fixed point where $g' = 0$, i.e., $7 - \frac{7}{4}g = 0 \Rightarrow g = 4$). Since $f(1)\neq 4$, the limit still converges to $\mathbf{4}$.