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Recently, I just found myself calling limits that can be solved using algebra as "intermediate limits". Example:

$$ \lim_{x \to 1} \frac{x^2-1}{x-1}$$

The expression evaluates to an indeterminate form.

But, if we factor out:

$$ \lim_{x \to 1} \frac{(x-1)(x+1)}{x-1} = \lim_{x \to 1} {(x+1)} = 2$$

Intermediate limit: limit that evaluate to an indeterminate form but can be simplified with algebra, to a limit which evaluates to a number.

The reason why I call them "intermediate limits", is that they're not so easy as so you can just evaluate, but they're not so hard they can't be solvable with just plain algebra. Hence, the "intermediate".

Question: Is it common notion? And is it misleading or correct?

Eric Wofsey
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    Not that I know of. Personally, I would call them "easy", but the point is that you simply have a predilection for a certain class of continuous functions: the functions $G(x)=\begin{cases}\frac{\sin x}x&\text{if }x\ne0\ 1&\text{if }x=0\end{cases}$ and $\lvert x\rvert$ are is just as continuous at $0$ as $x+2$, but you prefer making an argument for proving the others fact. –  Jan 05 '21 at 15:54

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I've never heard the term 'intermediate limit' before, but the limit you used an example is known as indeterminate form. An indeterminate form is where you don't have enough information to compute a limit just by looking at the individual terms. For example, $0/0$ is an indeterminate form, because $$ \lim_{x \to c}\frac{f(x)}{g(x)} $$ cannot be evaluated simply with the knowledge that $f(x) \to 0$ and $g(x) \to 0$. Compare this with $$ \lim_{x \to c}\frac{u(x)}{v(x)} $$ where $u(x) \to 5$ and $v(x) \to 3$ (as $x$ tends towards $c$). This limit is easy to solve because we can use the quotient law: $$ \lim_{x \to c}\frac{u(x)}{v(x)} = \frac{\lim_{x \to c}u(x)}{\lim_{x \to c}v(x)}=\frac{5}{3} \, . $$ Regarding the limit you gave as an example, $$ \lim_{x \to 1}\frac{x^2-1}{x-1} \, , $$ you might find this thread to be of interest: 'Why does factoring eliminate a hole in limit?'.

Joe
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  • "intermediate", not "indeterminate" –  Jan 05 '21 at 15:58
  • @1613585 I misread your question, sorry. However, I have never heard the phrase 'intermediate limit' either. I still think that much of my answer bears relevance to your question, but if not, you can ask me any questions in the comments. – Joe Jan 05 '21 at 16:00