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For each $t\in\mathbb{R}$, let $[t]$ denote the greatest integer $\leq t$. Evaluate: $\displaystyle\lim_{x\to1^+}\dfrac{(1-|x|+\sin|1-x|)\sin\!\left(\dfrac{\pi}{2}[1-x]\right)}{|1-x|\cdot[1-x]}$
As $x\to1^+$: let $h=x-1>0$, so $h\to0^+$. Then $|1-x|=h$, $[1-x]=[-h]=-1$.
$\sin\!\left(\dfrac{\pi}{2}[-h]\right)=\sin\!\left(\dfrac{\pi}{2}\cdot(-1)\right)=\sin(-\pi/2)=-1$
Numerator: $(1-|x|+\sin h)\cdot(-1)$. As $x\to1^+$: $|x|=x=1+h$, so $1-|x|=-h$.
Numerator $= (-h+\sin h)(-1) = h-\sin h$
Denominator: $h\cdot(-1)=-h$
Limit $= \dfrac{h-\sin h}{-h} = -(1-\dfrac{\sin h}{h})\to-(1-1)=\mathbf{0}$
The corpus luteum is primarily formed from which structure after ovulation?
After ovulation, the ruptured Graafian follicle collapses and transforms into the corpus luteum under LH influence. The ovum is released; oogonium and oocytes are germ cells.
Compare the two quantities below.
Column A: $(0.82)^2 \cdot (0.82)^3$
Column B: $(0.82)^6$
Using the rule of exponents: $(0.82)^2 \cdot (0.82)^3 = (0.82)^{2+3} = (0.82)^5$.
Now compare $(0.82)^5$ vs $(0.82)^6$.
Since $0 < 0.82 < 1$, multiplying by $0.82$ makes the number smaller. Therefore $(0.82)^5 > (0.82)^6$.
Column A is greater.
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