Indeterminate form of limit: is it obligatory to write it?

Question

I'm preparing for a Limits/Derivatives exam.

I have two very basic questions that I couldn't find an answer to.

Question 1: when calculating limit of a sequence or of a function, do I always have to write down the indeterminate form? If I solve the limit without writing down the indeterminate form, would it be incorrect (not full answer)?

Question 2: I am familiar with all seven indeterminate forms, however would it be correct to write it down like that? Example:

$$ \lim \frac{n}{n-1} = \bigg[ \frac{\infty}{\infty -1} \bigg] = \bigg[ \frac{\infty}{\infty} \bigg] = \text{. . . rest of solution . . .}$$

Of course I could write down the $\bigg[ \frac{\infty}{\infty} \bigg]$ right away. I came up with this simple example on purpose. $ \bigg[ \frac{\infty}{\infty -1} \bigg] $ - just need to know if such symbol is correct to use.

So essentially I need to know whether it's okay to "do calculations on the indeterminate form, inside the square brackets".

Thanks!

Answer 1

None of us can answer on behalf of your teacher/lecturer, but here comes my personal take on your questions:

Q1: I would say it is always good practice to state that the given limit is of one of the indeterminate forms (if this is the case!). I would say that probably 99% of the questions about limits ever given out at exams are about indeterminate forms - this is just because teachers want to 'test' students' knowledge on 'difficult' cases. However, it is not impossible that the teacher would put one or more non-indeterminate forms in their exam, just to test that students are able to spot them and act accordingly! It is not unheard of to have students who apply L'Hopital rule without thinking twice to limits $\lim_{x\to x_0} f(x)$ that are simply equal to $f(x_0)$ by continuity (no indeterminate form whatsoever!). Therefore, I would say it is always good to check at the very beginning of the calculation that you are truly facing an indeterminate form, and state so explicitly.

Q2: I do not have experience of the symbol $\lbrack \frac{\infty}{\infty -1} \rbrack$ been ever used. You can just state in words that 'this limit is in the indeterminate form $[\frac{\infty}{\infty}]$'. What you should probably not do (in general) is to continue writing your solution by putting an equal sign after $[\frac{\infty}{\infty}]$, that is to say: it is not OK in general to keep working inside square brackets. Once you know it is an indeterminate form, that's when you need to start working on the function and perform algebraic manipulations on it - I don't see how you can do that on the symbolic expression $[\frac{\infty}{\infty}]$!