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I have a question regarding the following limit calculation:

$\lim_{x \to 0} \frac{\cos(x)}{\sin(x)}$

The only way I can solve this is by looking at the one-sided limits:

$\\$ From above: $\lim_{x \to 0^{+}} \frac{\cos(x)}{\sin(x)}$.

The numerator approaches $1$ with a positive sign. The denominator approaches $0$ with a positive sign. $\implies$ the limit is $\infty$

$\\$ From below: $\lim_{x \to 0^{-}} \frac{\cos(x)}{\sin(x)}$.

The numerator approaches $1$ with a positive sign. The denominator approaches $0$ with a negative sign. $\implies$ the limit is $-\infty$

The one-sided limits do not agree and so the limit does not exist.

My concern is this: would you give full marks for an answer like this? It feels very informal but I do not know how to argue the same thing formally.

N. F. Taussig
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Hector Auvinen
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    I'm no teacher, but if this is a calculus class and not a real analysis class, your rationale is sound (and shows understanding). Have you learned the epsilon-delta definition of limit yet? If so, you can show that, for any L, the right side can be made larger than L when x is sufficiently close to 0, and analogously from the left side, less than L, a contradiction. (Comment if you want a formal expansion of that proof.) – Ovinus Real Jun 27 '22 at 19:12
  • What’s the difference between calculus and real analysis @OvinusReal? Isn’t analysis just advanced and complicated calculus using sup, inf and epsilon-deltas? I mean, I haven’t studied analysis, so I’d like to know – insipidintegrator Jun 27 '22 at 19:29
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    +1: to your question. Nice work shown, independent of whether you reached the right conclusion. For what it's worth, I (also) completely agree with your work. – user2661923 Jun 27 '22 at 19:31
  • Your explanation is OK for a high school student. If you are a college student, I would expect you to use epsilon-delta proof for one-sided limits. – Vasili Jun 27 '22 at 19:41
  • @insipidintegrator There's a bit more to real analysis, more fundamental things like how we construct the real numbers and the topology of $\Bbb R$, as well as more general notions of convergence (e.g. uniform). This page has a nice overview of topics. – user170231 Jun 27 '22 at 19:56
  • @algevristis The limit is not $0$, it is infinite - either positive or negative. You can't use L'Hospital's rule here as it's not a $\frac00$ or $\frac\infty\infty$ form. – PC1 Jun 27 '22 at 20:05

1 Answers1

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I would give you full score without any hesitation. Some professors would ask for an $\epsilon$-$\delta$ proof but in my opinion this would be counter productive. Here are some reasons why I like your reasoning:

  • You paired the problem to its basic, each sub-goal you have is easy, then you just have to assemble the pieces together and you did tell how you do it.

  • Computing the limit of all the terms that are in your expression and then saying that taking a fraction preserves the limit (when the limiting numbers are fractions-friendly) is way more natural than guessing the limit (with your process!) and parachute an $\epsilon$-$\delta$ proof making it "formal".

However if you have any doubt about your proof then you should be more precise and rigorous, which does not necessarily mean performing an $\epsilon$-$\delta$ proof. For instance if you do not trust the tools you used: prove them. Proving the two limit theorems you used is way better than parachuting an $\epsilon$-$\delta$ proof. For mathematics I think a key habit to learn something is to prove everything until you are convinced and without any remaining doubt.

Disclaimer: this is my opinion and I am not a teacher/professor, only a student between first and second year of Master of mathematics.

blamethelag
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