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My teacher is a stickler for correct notation. When evaluating a limit, when do you stop using the limit notation $\lim \limits_{x\to c}\left(f(x)\right)$ and just write a value?

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When you have a value. You use the limit notation until you’ve actually evaluated the limit (though there may be a few more steps in which you simplify your expression for that value, e.g., converting $\frac{3+1}6$ to $\frac23$). – Brian M. Scott Sep 18 '13 at 21:53
I'd say that your teacher is a "stickler" for understanding the material he teaches. – Vedran Šego Sep 18 '13 at 22:11

You stop using it if and when you actually evaluate the limit, i.e. in circumstances where a limit exists and is finite, and you are recording its initial evaluation $L$. $$\lim_{x\to c} f(x) = L$$

If the limit does not exist, you can write something to the effect $$ \lim_{x\to c} f(x) \;\text{does not exist.}$$

If the limit diverges to (+) infinity, e.g., you might write something to the effect $$\lim_{x\to c} f(x) \; \text{diverges to } \infty$$ though in the divergent case, when the limit diverges to (+) infinity, e.g., you'll sometimes see $\lim_{x\to c} f(x) \to \infty,\;$ or $\;$"as $x\to c$, $f(x) \to \infty$." Follow your instructor's lead here, in this scenario.

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1. If you've proven that a limit exists, I see no problem in using a variable to refer to it, even if you haven't evaluated it numerically. 2. And on the other hand, why not continue using the $\lim$ symbol instead of burning another letter to define a new variable? – Jack M Sep 18 '13 at 22:12
Some students seem to be lazy and don't like writing $\lim$ over and over again, even though sometimes it needs to be done. – Stefan Smith Sep 19 '13 at 1:50

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