1. ## Help Needed with Advanced Limits # 5

I was wondering if someone could help me with the problems within the picture:

I am in dire need of a step by step explanation as I answered this as 3/4 and was incorrect

2. I wouldn't be surprised if the book is wrong, considering that they can't even use latex correctly.

3. really?? I figured that if F(x)=3x and G(x)=4x then my answer would make sense, but my professor says otherwise...

4. Do you know L'Hopital rule?
$\displaystyle \lim_{x \to p} \frac {f(x)}{g(x)}=\lim_{x \to p} \frac {f'(x)}{g'(x)}$

5. oh right......man how did I miss that, lol. Yeah of course I know about L'Hopital's and so now my answer really makes sense. But just to be sure, is there no possible way for me to be incorrect?? What about if F(x)/G(x) isn't 0/0 of INFINITY/INFINITY??

6. Can anyone verify this?? Is the answer really 3/4?? My professor seems to think otherwise and he seems to always be right. Any clarification of verification is greatly appreciated

7. Originally Posted by Some1Godlier
Can anyone verify this?? Is the answer really 3/4?? My professor seems to think otherwise and he seems to always be right. Any clarification of verification is greatly appreciated
The question is either absurd or incomplete (see below) because as posted there is no way of knowing $\displaystyle \lim_{x \rightarrow p} \frac{f(x)}{g(x)}$.

eg. If $\displaystyle f'(x) = 3x$, $\displaystyle g'(x) = 4x^2$ and $\displaystyle p = 1$ then it should be clear that $\displaystyle \lim_{x \rightarrow 1} \frac{f'(x)}{g'(x)} = \frac{3}{4}$. However:

$\displaystyle \lim_{x \rightarrow 1} \frac{f(x)}{g(x)} = \lim_{x \rightarrow 1} \frac{\frac{3}{2}x^2 + A}{\frac{4}{3} x^3 + B} = \frac{\frac{3}{2} + A}{\frac{4}{3} + B} = \frac{9 + C}{8 + D}$

where A and B and hence C and D are completely arbitrary.

The missing information is most likely the fact that either $\displaystyle \lim_{x \rightarrow p} f(x) = \lim_{x \rightarrow p} g(x) = \pm \infty$ OR $\displaystyle \lim_{x \rightarrow p} f(x) = \lim_{x \rightarrow p} g(x) = 0$. In which case it should be clear from l'Hopital's rule that the answer would indeed be 3/4.

Perhaps you should discuss all this with your professor who "seems to always be right" ....