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The limit of (x-2) - x as x approaches infinity is 2 (the xs cancel and you’re just left with 2 in the limit expression), so even as you go towards infinity the distance between the lines stays two.
imagine two intersecting lines. now start tilting one so that the intersection moves away. as you tilt the line more and more, the lines approach parallelism and the intersection goes to infinity
Even though the lines are infinitely far away from the x axis, they are still the same distance apart. Even if you parameterizing the slope of one line and modify it:
Ax - 2 = x
Intersection is
Ax - 2 - x = 0
x = 2 / (A - 1)
So they become parallel as A approaches 1
So the inter section point for A = 1 is
lim 2 / (A-1) as A -> 1
Which is 2/0, or undefined.
You can gain a lot of insight with graphical explanations, but sometimes can fall into traps, so it’s helpful to translate that to formal notation sometimes.
So like, in any real system as x approaches infinity the difference between x and x+2 will fall below measurement error and make x = x+2 functionally true (far field small angle approximation and whatnot.) This kind of situation comes up in optics when you’re finding different f points on lenses. I think it’s more a case of “both ways to consider that math are useful in different regimes/circumstances.” If a student in a proof based math class i was teaching came at me with the graphical explanation I’d tell them to try again, but if one of my junior engineers came to my office with concerns about the difference between x and x+2 at x=50000 or something id take it as an opportunity to teach them why it probably doesn’t matter.
The limit of (x-2) - x as x approaches infinity is 2 (the xs cancel and you’re just left with 2 in the limit expression), so even as you go towards infinity the distance between the lines stays two.
imagine two intersecting lines. now start tilting one so that the intersection moves away. as you tilt the line more and more, the lines approach parallelism and the intersection goes to infinity
Even though the lines are infinitely far away from the x axis, they are still the same distance apart. Even if you parameterizing the slope of one line and modify it:
Ax - 2 = x Intersection is Ax - 2 - x = 0 x = 2 / (A - 1)
So they become parallel as A approaches 1
So the inter section point for A = 1 is lim 2 / (A-1) as A -> 1
Which is 2/0, or undefined.
You can gain a lot of insight with graphical explanations, but sometimes can fall into traps, so it’s helpful to translate that to formal notation sometimes.
So like, in any real system as x approaches infinity the difference between x and x+2 will fall below measurement error and make x = x+2 functionally true (far field small angle approximation and whatnot.) This kind of situation comes up in optics when you’re finding different f points on lenses. I think it’s more a case of “both ways to consider that math are useful in different regimes/circumstances.” If a student in a proof based math class i was teaching came at me with the graphical explanation I’d tell them to try again, but if one of my junior engineers came to my office with concerns about the difference between x and x+2 at x=50000 or something id take it as an opportunity to teach them why it probably doesn’t matter.
Oh, in an engineering context, yeah. I think that’s more because you’re looking at the relative difference between the two, or
((x-2)-x)/x, which is -2/x, which is 0 as x approaches infinity.
I can do equations for you:
y = ax
y = x+2
with a>1. Intersection at x=2/(a-1). As a goes to 1, first line approaches y=x and the intersection point goes to infinity.