A quick asymptote question, 10 points best answer?

Question

Outta curiousity, asymptotes are values that a curve approaches, but never meet the curve at those lines, but in curves like

f(x) = 1 + (x / x^2 +1)


they cross the asymptote heaps of times xD, why's that?

The horizontal asymptote is y= 1, and yet thats its axis intercept...

Is there a special case with horizontal asymptotes?

And can a curve cross its verticle asymptotes?

I guarentee there is some retarded curve out there xD But why can it cross, they aint meant to touch each! =S

Answer 1

This function is not defined at zero -- so it has no intercept.

That being said, your question is still valid. However, if you look closely at what it means to be an asymptote, you will see that the definition only describes what happens at extremes. For example, with this function we only care about what happens when X gets large. For large x, this function is monotonically decreasing.

A more interesting example would be something like

f(x) = 1+(sin(x)/x)

This function keeps crossing the asymptote over and over. But the limit as x gets large still goes to one. We can show this by proving that this function is bounded by an increasing function (1-1/x) and a decreasing function (1+1/x) which both have an asymptote of one. Therefore, the linit of f(x) as x goes to infinity must also be one.

Answer 2

initially you're finding for the type this is the y values not the x values. you may not sq. root a adverse quantity or 0 so the area (the x-values) is (0, infinity) put in a quantity on the brink of 0 say .001 then you certainly've the sqrt(a million.001)/sqrt(.001) = a million.0005/.0316 this is approximately 31.6 b/c you're dividing a quantity on the brink of a million by potential of a smaller quantity this might consequence in a great answer. try some thing even smaller like .0000005 and you will see the consequence is approximately 1414 Now plug in a great quantity working example ninety 9 then you certainly've the sqrt(a hundred)/sqrt(ninety 9)this is a million.0.5 because of the fact the numbers are on the brink of another they only approximately have a ratio of a million yet by no potential would be a million precisely because of the fact the numerator will consistently be slightly extra desirable than the denominator. which potential the type is (a million, infinity) i'm hoping this helps.

Answer 3

Your answer y=1 is correct.
f(x)=1/(x-1)
x=1 is the vertical asymptote. Did you mean whether the horizontal asymptote can cross the vertical asymptote? To the best of my knowledge, the answer is no. The curve cannot cross its vertical asymptotes.

Answer 4

f(0) = 1
f(inf) = 1