i was wondering if somebody had any ideas about this:

 

the gradient of a horizontal line = 0

the gradient of a vertical line = infinity

 

but the product of the gradients of perpendicular lines is -1

 

so does that mean that 0 x infinity = -1 ???

 

... or is there something in the proof for the product of the gradients equalling -1 which excludes it for these values?

The limit of x->infinity multiplied by limit of y->0 from below is -1, but the gradient of a vertical line is undefined.

but the product of the gradients of perpendicular lines is -1

 

 

it is? but that would require infinity to be a real number, which it isn't. (that is to say this result is only true if both gradients are defined, and infinity is not acceptable in this context)