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I am currently reviewing an old Calculus textbook and I stumbled upon two questions that, for me, has the wrong answer on the answer key. I would appreciate if you could check if my reasoning is correct. They follow:

Question 1) If $\lim_{x \to 5} f(x) = 2$ and $\lim_{x \to 5} g(x) = 0$ then $\lim_{x \to 5} \frac{f(x)}{g(x)}$ does not exist. True or false?

Textbook's answer: True

My answer: False (isn't that what L'Hôpital is all about?)

Question 2) If $\lim_{x \to 1} f(x) = 4$ and $\lim_{x \to a}$ does not exist then $\lim_{x \to a} \left[ f(x) + g(x) \right]$ does not exist

Textbook's answer: True

For this second one, I agree but just wanted to make sure if the textbook's answer is correct.

EDIT: I see my mistake on the top one: the upper limit of $f(x)$ is $2$ not $0$ (I've read it too fast!)

Thank you.

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    $\begingroup$ 1 is indeed True, L'Hospital's rule applies only to indeterminate forms like $\frac{0}{0}$ and $\frac{\infty}{\infty}$. So just like you edited. The second assertion is correct too indeed. $\endgroup$ – Václav Mordvinov Mar 10 '18 at 14:43
  • $\begingroup$ If you think the first answer is False, find please a limit $\lim_{x \to 5} \frac{2}{x-5}$ $\endgroup$ – D F Mar 10 '18 at 14:44
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Question 1: The statement is true: the limit does not exist (in $\mathbb R$, at least; it could be $\pm\infty$). I don't understand your reference to L'Hopital's Rule.

Question 2: Yes, the statement is true.

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The book is right both times. You are wrong on the first.

This has nothing to do with L'Hopital. You're not told anything about differentiability. And the book would be right even if $f$ and $g$ were differentiable.

Personal note: "L'Hopital's rule" is much abused in elementary calculus. It's often invoked when there are simpler and more intuitive ways to find a limit, and it's often invoked without checking the assumptions that make it applicable.

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