Hmmmm......
See initially when reciprocating graphs I got taught that as y --> infinity, then 1/y ---> 0 and therefore the x-intercepts of the reciprocal graph occur when there is an asymptote on the y-axis. But then the problem that arises is that can we consider infinity as a number (without applying limits) which has the property of 1/infinity = 0?
For many of the reciprocal graphs I've drawn, y = 0 is an asymptote. Eg for y =x^2 where the reciprocal is just the basic truncus. While y = x^2 doesn't have an asymptote, as x--->infinity, y ---> infinity and hence for the reciprocal graph we say y----> 0 and that y = 0 is the asymptote. Here there is no crossing the asymptote and it is pretty clear that no matter what y will not equal zero. So if we are sketching the reciprocal of tan x, or even things like log x, why should they have x-intercepts at the asymptote?? This confuses me a bit.
As for limits, I think that can be deceptive because even if we have y = x, x is an element of R\{2}. Then the limits would tell us that as x ---> 2, then y ---> 2, but we would still put an open circle there because we know that x can't actually = 2. I thought limits in such cases were mainly to test differentiability but then it's just the start of the spesh course and I don't know half the stuff taught in spesh yet (let alone uni stuff which some people on this forum have covered).
Home
Help
Search
Login
