horizontal asymptotes can be crossed...

[taiga]

The precise definition of a horizontal asymptote is the following:

Let be defined on then means for every there exists a corresponding number such that if then . Likewise let be defined on then means for every there is a corresponding such that if then .

is the horizontal asymptote.

Thus horizontal asymptote can certainly be crossed.


Exercise for those interested: Using the same logic as above, can you formalise vertical asymptotes?

but in all honestly no one read that

[8039]

The precise definition of a horizontal asymptote is the following:

Let be defined on then means for every there exists a corresponding number such that if then . Likewise let be defined on then means for every there is a corresponding such that if then .

is the horizontal asymptote.

Thus horizontal asymptote can certainly be crossed.


Exercise for those interested: Using the same logic as above, can you formalise vertical asymptotes?

but in all honestly no one read that

I read it but in the same way I read Shakespeare... not knowing wtf is going on

[TrueTears]

the mathematical language used is within methods boundaries and i bet if i explained it a bit more using english everyone wud understand, it's actually very very intuitive except i cbf lol.. maybe kamil or someone can translate into english for ya guys

[8039]

the mathematical language used is within methods boundaries and i bet if i explained it a bit more using english everyone wud understand, it's actually very very intuitive except i cbf lol..

Well I had enough trouble understanding limits in the Essential book so that explains my confusion. I kinda get it it till x>N and the absolute value thing thingo. And how it proves that the horizontal line can be crossed

[Martoman]

Well TT sir, i prefer good old english.

When you have something like if you let y = 0 then you can have either

or

For the first one, this cannot ever happen, but it *can* for the second. It is asymptotic for extremely small values of x because so it approaches zero as x becomes extremely small.

But for things like

if you let y = 7 then you have the unsavoury situation of implying and o noooooo the universe has sufficiently exploded. ie proof by contradiction.

This is why you cannot cross vertical ones. You will invariably end up with some ridiculous statement like 1= 9 or a = b where a doesn't = b....  :uglystupid2:

For the question

On that same topic, I wonder whether the graph has an asymptote

e^-x = 0, sin(x) = 0

x has a solution at x = 0, y = 0 so it does cross, yet behaves asymptotically because as you tend towards infinity, so it appraoches 0

[TrueTears]

yer in vce maths u dont have to be so formal lol (its coz im reading a few books on elementary topology and analysis so i'm being a formal bastard with all the maths i do nowadays LOL) but once you go past vce, being formal with these asymptotes are great! You can do wonderful proofs and stuff~

[8039]

Well TT sir, i prefer good old english.

When you have something like if you let y = 0 then you can have either

or

For the first one, this cannot ever happen, but it *can* for the second. It is asymptotic for extremely small values of x because so it approaches zero as x becomes extremely small.

But for things like

if you let y = 7 then you have the unsavoury situation of implying and o noooooo the universe has sufficiently exploded. ie proof by contradiction.

This is why you cannot cross vertical ones. You will invariably end up with some ridiculous statement like 1= 9 or a = b where a doesn't = b....  :uglystupid2:

For the question

On that same topic, I wonder whether the graph has an asymptote

e^-x = 0, sin(x) = 0

x has a solution at x = 0, y = 0 so it does cross, yet behaves asymptotically because as you tend towards infinity, so it appraoches 0

"Sorry, you can't repeat a karma action without waiting 12 hours."

[/0]

The precise definition of a horizontal asymptote is the following:

Let be defined on then means for every there exists a corresponding number such that if then . Likewise let be defined on then means for every there is a corresponding such that if then .

is the horizontal asymptote.

Thus horizontal asymptote can certainly be crossed.


Exercise for those interested: Using the same logic as above, can you formalise vertical asymptotes?

Reading that hurt my head -_-

lol that's basically all of Analysis

[TrueTears]

analysis!!!!!

[Martoman]

analysis!!!!!