# Two tricky limit questions regarding formal definition (possible textbook mistake)

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?

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

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.

• 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. – Václav Mordvinov Mar 10 '18 at 14:43
• If you think the first answer is False, find please a limit $\lim_{x \to 5} \frac{2}{x-5}$ – D F Mar 10 '18 at 14:44

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.
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.