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Which cytoplasmic organelle is capable of synthesizing its own proteins?
Mitochondria have their own DNA (mtDNA) and ribosomes (70S) and can synthesize some of their proteins. Chloroplasts also do this.
For each $x\in\mathbb{R}$, let $[x]$ be the greatest integer $\leq x$. Find: $\displaystyle\lim_{x\to0^-}\dfrac{x([x]+|x|)\sin[x]}{|x|}$
For $x\to0^-$: $x<0$, $|x|=-x$, $[x]=-1$.
$\dfrac{x((-1)+(-x))\sin(-1)}{-x} = \dfrac{x(-1-x)(-\sin1)}{-x}$
$= \dfrac{x(1+x)\sin1}{x} = (1+x)\sin1$
As $x\to0^-$: limit $= 1\cdot\sin1$... wait: $= \dfrac{x(-1-x)(-\sin1)}{-x} = \dfrac{(1+x)\sin1 \cdot x}{x}=(1+x)\sin1\to\sin1$? Re-doing: numerator $= x(-1-x)\sin(-1)=x(-1-x)(-\sin1)=x(1+x)\sin1$; denominator $=-x$. So $=\dfrac{x(1+x)\sin1}{-x}=-(1+x)\sin1\to-\sin1$. Answer: $\mathbf{-\sin 1}$.
Who developed the 'Greatest happiness of the greatest number' principle in Utilitarianism?
Jeremy Bentham is the founder of the Utilitarian school of thought.
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