True or False?
a) If $f(x) → 0$ as $x → a^+$,(from the right) and $g(x) \ge 1$ for all $x$ in $\Bbb R$, then $g(x)/f(x) → ∞$ as $x → a^+$.
True: take $f(x) = \sin x$ and $g(x) = x^2$ as $x → pi/2$ from the right. Then the limit is $1/\infty$ thus approaches zero.
Is this correct. Please can someone please verify this?
Thank you.
The limit $g(x)/f(x)$ diverges, but you cannot say the sign (e.g. $g(x)=1$, $a=0$, and test with $f(x)=x$ and $f(x)=-x$).
Moreover you have $$\lim_{x\rightarrow \pi/2} \frac{x^2}{\sin x}=\frac{\pi^2}{4}$$ since $\sin (\pi/2)=1$.