As $x\to 0^+$, $x$ is always positive, but it’s getting smaller and smaller. Consequently, $\frac1x$ is also positive, but it’s getting bigger and bigger, and $\frac1x\to\infty$. It would probably help you to draw a graph of $y=\frac1x$, or display it on a graphing calculator: you’d see that as $x$ moves leftwards towards $0$, $\frac1x$ shoots rapidly upwards. There’s a picture here.
Now take it a step further. Let $y=\frac1x$, and ask yourself what happens to $\tan^{-1}y$ as $y\to\infty$. Recall that $\tan^{-1}y$ is the angle (in the first or fourth quadrant) whose tangent is $y$; what kinds of angles have very large tangents? Let’s see: $\tan\frac{\pi}6=\frac12,\tan\frac{\pi}4=1$, and $\tan\frac{\pi}3=\sqrt3\approx1.732$, to take a few familiar angles, so the tangent seems to be getting bigger as the angle increases. In fact the tangent an angle in the first quadrant is simply the slope of the line through the origin that makes that angle with the positive $x$-axis. As that slope gets bigger and bigger, what’s happening to the angle between the line and the $x$-axis? It’s approaching a right angle, or $\frac{\pi}2$.