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.

Question

amistre64 · Answer

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

anonymous · Answer

No as it is

anonymous · Answer

I mean using the math rule

anonymous · Answer

First division then addition...................

amistre64 · Answer

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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

Yes

anonymous · Answer

$$y==1+\frac{1}{x} $$
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.

anonymous · Answer

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

anonymous · Answer

Thanks robtobey

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

And I have presented the problem that I face while reasoning

anonymous · Answer

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

anonymous · Answer

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

amistre64 · Answer

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

anonymous · Answer

Yes sure

amistre64 · Answer

then the asymptote is a slant/oblique right?

anonymous · Answer

Sorry, but how did you conclude that?

amistre64 · Answer

the ends approach y=x

anonymous · Answer

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

amistre64 · Answer

the top is one degree highr than the bottom

anonymous · Answer

Actually I am not aware of that rule

amistre64 · Answer

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

anonymous · Answer

Yes

amistre64 · Answer

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

anonymous · Answer

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

anonymous · Answer

Is that what you mean?

amistre64 · Answer

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

anonymous · Answer

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

amistre64 · Answer

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

amistre64 · Answer

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

amistre64 · Answer

whats left over is what your asymptote is :)

anonymous · Answer

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

amistre64 · Answer

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)