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
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" ....