Calculating limit

Calculating limit

I need confirmation for the following question.
Which of the following statements is true?
$1.$ $\lim_{x\to\infty}\frac{\log x}{x^{1/2}}=0$ and
$\lim_{x\to\infty}\frac{\log x}x=\infty$
$2.$ $\lim_{x\to\infty}\frac{\log x}{x^{1/2}}=\infty$ and
$\lim_{x\to\infty}\frac{\log x}x=0$
$3.$ $\lim_{x\to\infty}\frac{\log x}{x^{1/2}}=0$ and
$\lim_{x\to\infty}\frac{\log x}x=0$
$4.$ $\lim_{x\to\infty}\frac{\log x}{x^{1/2}}=0$ but
$\lim_{x\to\infty}\frac{\log x}x$ does not exist.
So, I am thinking that however large $x$ be, $\log x$ will remain small.
So, a small quantity over a very large one would give us $0$ as result.
And sqrt of a large number is also a large number. So, answer should be
$3).$
Am I correct here?