Where $\phi(z)$ and $\Phi(z)$ represent the standard normal pdf and cdf respectively.
1) Is the function
$$f(z)=\frac{\phi(z)}{1-\Phi(z)}$$
increasing for all values of $z$? If so, how can I show it?
2) Is the limit as $z\rightarrow\infty$ using L'Hôpital's rule
$$\lim_{z\rightarrow\infty}f(z) = \frac{\phi'(z)}{-\phi(z)}=\frac{-z\phi(z)}{-\phi(z)}=z=\infty \text{?}$$
The ratio yoy have given, is known as the inverse Mills ratio, see: https://en.wikipedia.org/wiki/Inverse_Mills_ratio From that article, we have the representation, if $X$ is a random variable with the standard normal distribution (expectation zero, variance one) that $$ E(X | x > \alpha) = \frac{\phi(\alpha)}{1-\Phi(\alpha)} = f(\alpha) $$ and from that interpretation it follows that $f$ is indeed increasing. You can also try to prove it directly using the usual differentiation rules.
As for your question 2) I think you solved it correctly.
