[kenbellows]

1/x as x -> 0 is a famously tricky expression because your statement that it "obviously increases exponentially as x approaches zero" is only true half the time, because it's only true if you approach zero from the positive side, with x being set to smaller and smaller positive values. If, on the other hand, you set x to negative values of smaller and smaller size, 1/x actually decreases toward -Infinity. This is one of the several reasons that 1/x is typically considered undefined for the Reals.

[mirimir]

Damn, so last night I was thinking about how 1/x approaches ∞ for positive x, and -∞ for negative x. And it struck me that perhaps beyond both ∞ and -∞ is 0, so 1/x might actually equal 0 when x is 0. Rather like closed "liner" paths in closed universes. But again, IANAM ;)

[mirimir]

OK, sure. Either infinity or -infinity. But then just look at the absolute value of 1/x. And in any case, neither infinity nor -infinity look anything like zero.