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Let $[x]$ denote the greatest integer $\leq x$. Find: $\displaystyle\lim_{x\to0}\dfrac{\tan(\pi\sin^2 x)+(|x|-\sin(x[x]))^2}{x^2}$
Split into right and left limits.
Right limit ($x\to0^+$): $[x]=0$, so $\sin(x[x])=0$, $|x|=x$.
$\dfrac{\tan(\pi\sin^2x)+x^2}{x^2}$. Using $\tan(\pi\sin^2x)\approx\pi\sin^2x\approx\pi x^2$:
$=\pi+1$
Left limit ($x\to0^-$): $[x]=-1$, $\sin(x[x])=\sin(-x)=-\sin x$, $|x|=-x$.
$=\dfrac{\tan(\pi\sin^2x)+(-x-(-\sin x))^2}{x^2}=\dfrac{\pi x^2+(\sin x-x)^2}{x^2}\to\pi+0=\pi$
Left $\neq$ Right, so the limit does not exist.
Factory data: Budgeted production 10,000 units; Budgeted overheads $50,000; Actual overheads $60,000; Under-absorption $5,000. What was the actual production?
Option A (9,000 units) is correct.
OAR = $50,000 ÷ 10,000 = $5 per unit
Under-absorption = Actual overheads − Absorbed overheads
$5,000 = $60,000 − (Actual units × $5)
Actual units = $55,000 ÷ $5 = 9,000 units
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