Why doesn't $f'(x) \to 0$ at $x \to \infty$ imply that $\lim_{x \to \infty}{f(x)}$ exists?

Slope going to $0$ means adding ever smaller increments to the function value over each interval of unit length, from $1$ to $2$, from $2$ to $3$, and so on. But even when the increments go to $0$ they can accumulate so that their sum grows without bound. It is the same effect as with harmonic series diverging. In fact, partial sums of harmonic series grow roughly at the same rate as $\ln x$ over unit intervals, $\ln(n+1)-\ln n \sim \frac1n$.

A neat physical illustration of this is that one can stack bricks on top of each other so that the stack is stable, but the top brick hangs over the bottom one as far horizontally as one wishes. Each brick has to be shifted sideways from the previous one by $\frac1{2n}$ of its length, see Overhang puzzle. This is even more counterintuitive than a function growing to $\infty$ while its tangent approaches the horizontal.

Does it seem "intuitively obvious" to you that, if my velocity approaches zero, then my position must approach a limiting value? (Velocity is the derivative of position.) Let me get in my spaceship for this, I'm going to need lots of room. Here's my itinerary.

First I fly $1$ mile north at $1$ mile per hour.
Then I fly $10$ miles south at $0.1$ mph.
Then $100$ miles north at $0.01$ mph.
Then $1000$ miles south at $0.001$ mph.
Then $10000$ miles north at $0.0001$ mph.
And so on, ad infinitum.

Graph my position as a function of time. Does the slope approach zero? What limit does the function converge to?