## anonymous 5 years ago I am trying to sketch the curve y=x+1/x Following are the steps that are generally followed. But I don’t understand one thing First I note that while as x tends to infinity, 1/x tends to 0. Then we conclude that y tends x, since 1/x tends to 0. The thing that I don’t understand is if x tends to infinity, so why don't we say y tends to infinity too.

1. amistre64

y = (x+1)/x ? or y = x + (1/x)

2. anonymous

No as it is

3. anonymous

I mean using the math rule

4. anonymous

5. amistre64

id say its y = inf + 0 as well...... is there an answer that says differently?

6. anonymous

Its curve sketching. If I say that way, the curve turns out to be wrong

7. anonymous

iamignorant, Is the following your expression? $y=x+\frac{1}{x}$

8. anonymous

Yes

9. anonymous

$y==1+\frac{1}{x}$ The 2 attached plots should help visualize the function. The first plot is from x=-10 to +10. The scale of the plot is too large and should be reduced in your pdf viewer. The second is from x= -4000 to x= +4000. To the the casual observer the plot looks like a straight line drawn through the origin and making an angle of Pi/4 with the x axis.

10. anonymous

Just reviewed what was put on the net. The plots seem to be OK.

11. anonymous

Thanks robtobey

12. anonymous

You see the problem that is disturbing me is a little different

13. anonymous

I am trying to sketch the curve using the asymptote location method

14. anonymous

And I have presented the problem that I face while reasoning

15. anonymous

I understand that while 1/x tends to infinity y must tend to x, or else y=x will not be an asymptote

16. anonymous

But I don't understand where exactly I mis-reasoned to make it otherwise

17. amistre64

y = (x^2 +1)/x is an equivalent equation right?

18. anonymous

Yes sure

19. amistre64

then the asymptote is a slant/oblique right?

20. anonymous

Sorry, but how did you conclude that?

21. amistre64

the ends approach y=x

22. anonymous

Hold on a few minutes. I want to set up some Limits.

23. amistre64

the top is one degree highr than the bottom

24. anonymous

Actually I am not aware of that rule

25. amistre64

when we divide it all by x we get: x +1/x ------ 1

26. anonymous

Yes

27. amistre64

anyting with an x on the bottom goes to zero; which leaves us with y = x as the slant asymptote

28. anonymous

You mean when 1/x tends to zero anyting with an x on the bottom goes to zero

29. anonymous

Is that what you mean?

30. amistre64

so you have 2 curves that trail in the ends to the line y=x; and in the middle skirt along the x=0 axis: like this

31. anonymous

Good job amistre64. I have to break off. Thank you.

32. amistre64

when the bottom of a fraction gets large; the value gets small: 1 -------------- = .000000...00001 1000000...0000

33. amistre64

so anything with an x on the bottom; as the x gets large; the value goes to zero.

34. amistre64

whats left over is what your asymptote is :)

35. anonymous

Actually, when x is tending towards infinity 1/x tends to zero (this is what you say), but x is tending to infinity so y=x+0 means y is tending to infinity. This is what I don't understand

36. amistre64

y is keeps going up and skirting next to the y=x line; instead of a y=3 line, its the y=x line. that line has a range of (-inf,inf)