Limits. Thank You.
Look at it this way:
we use the limit theorem, If $\displaystyle \lim f(x) = F$ and $\displaystyle \lim g(x) = G$, then $\displaystyle \lim f(x)g(x) = \lim f(x) \cdot \lim g(x) = FG$
thus, $\displaystyle \lim_{t \to 0} \frac {e^t}{3t^2} = \lim_{t \to 0} \frac {e^t}3 \cdot \lim_{t \to 0} \frac 1{t^2}$
and of course, you can get really formal with this, using sequential approaches to limits, but let's not go there, since they didn't ask us to "prove"
short of doing a proof for this question, i'd do something like this (i hope it's valid ):
$\displaystyle \lim_{x \to 0^+} \frac {\ln x}x = \lim_{x \to 0^+} \frac 1x \cdot \lim_{x \to 0^+} \ln x$
since $\displaystyle \lim_{x \to 0^+} \frac 1x = \infty$ and $\displaystyle \lim_{x to 0^+} \ln x = - \infty$, that is, one is positive and the other is negative. we have that the products are negative and going to $\displaystyle - \infty$
thus $\displaystyle \lim_{x \to 0^+} \frac {\ln x}x = - \infty$
i know that's the answer, i don't know if what i did was 100% mathematically valid though, but it should be
TPH and Krizalid are the ones who are exceptional with limits