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As I'm working on my "derivatives of polynomial functions," and I'm learning about limits. This as goes well, except for the notation part. The first notation I encounter is
$$ \lim_{x \to a} f(x) $$ My textbook says I should read this as "limit for $x$ nearing $a$" or "limit in $a$." I am however not given any explanation as to why this is so, and how to adapt my reading to other formulas. So when I stumbled upon the next one...
$$ \lim_{h \to 0} 0=0 $$ ...I had no idea what this stood for. I'm guessing "the limit of $0$ with $h$ nearing $0$ is zero." But even if that's so, I've got no idea as to which zero stands for what and why. A bit of background info is that this last notation was the result of an example about derivatives with $y = 3$. The coordinates of the first point were $(1,3)$ and those of the second were $(1+h,3)$. The difference quotient was $0$.

If you could please explain how to read this notation and others of the like, and which part of the notation means what exactly (so that I can read others and know how to change my reading), that would be of great help!

Thanks in advance
Lila

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    $\begingroup$ Well, what textbook are you reading? $\endgroup$
    – Siminore
    Commented Apr 17, 2018 at 9:59
  • $\begingroup$ $0$ stands for $0.$ Suppose $f(h)=0$ for every $h$. Then $\lim_{h\to 0}f(h)=f(0)=0$. But every $f(h)$ IS 0 so we can replace $f(h)$ with $0$ wherever $f(h)$ appears $\endgroup$ Commented Apr 17, 2018 at 21:56

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The notation $$\lim_{x\to a}f(x)$$ is read as 'limit of the function $f(x)$ as $x$ approaches/moves near to/ tends to the number $a$.'

The $\lim$ is just shorthand for limit; $f(x)$ I hope you know; and the symbol $x\to a$ tells you that the variable $x$ gets closer and closer to the value $a$ (and sometimes $a$ can stand for $\pm\infty$). As time goes by you will get used to it, and maybe even get to see how it is a most natural notation for expressing a limit.

I hope this helps.

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