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Let $f$ be differentiable with $f'(x)=7-\dfrac{3}{4}\dfrac{f(x)}{x}$ for $x>0$ and $f(1)\neq 4$. Find $\displaystyle\lim_{x\to 0^+} x\,f\!\left(\dfrac{1}{x}\right)$.
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 $\mathbf{4}$.
To promote domestic energy production, a state offers a $5,000 tax credit to residents who buy solar panels manufactured within that state. No credit is available for panels bought from out-of-state. Is this constitutional?
Even if the state is spending its own money (subsidy), it cannot use its taxing power to discriminate against interstate commerce. Subsidies are often allowed under the 'Market Participant' exception, but discriminatory tax credits are almost always struck down.
The textbook states: “the earliest known printed text, the Diamond Sutra (a Buddhist scripture), was printed in China in 868 A.D.” It was created using block printing — a technique where individual pages are carved into wooden blocks and pressed onto paper. This text predates Gutenberg's press by nearly 600 years. The Diamond Sutra is a famous Mahayana Buddhist scripture, and the printed copy (now held in the British Library) is remarkable for its technical quality, indicating the technique had been refined over time before this particular copy was made. Note the other important dates: paper milling in Baghdad began as early as 794 AD; Bi Sheng invented movable type in 1041 AD; Gutenberg developed the European printing press in the 1440s; and the first printing press in Muslim territory (Andalusia) operated in the 1480s.
A trader sold goods worth $6,600 on 31 March 2025 but did not produce the invoice until 3 days later. The sale should be recorded in the year ended 31 March 2025. Which accounting concept applies?
- Consistency
- Prudence
- Realisation
Option D (3 only) is correct.
The realisation concept states revenue is recognised when earned (goods transferred and risks/rewards pass), not when the invoice is raised. The sale was complete on 31 March, so it belongs in that year.
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