How many asymptotes does the function have?

Question

f(x) = 1/x + ln(1+e^x)

Answer 1

Ok, 1/x has x and y axes as asymptotes and is basically a couple of parabolas in the top right and bottom left quadrants of the plane.

For negative x, 1+e^x tends to 1 as x tends to infinity, so the log of it tends to zero and the bottom left parabola stays roughly the same.

But for positive x, 1+e^x is roughly e^x and its log is therefore roughly x, so we have x +1/x which looks like 1/x to start with for x near zero, but then the x part takes over as x tends to infinity, so its a parabola in the wedge between the positive y axis and the positive part of the line y=x.

So the asymptotes are the positive and negative y-axis, the negative x-axis and the positive part of the line y=x. That's three or four asymptotes depending on whether you count the positive and negative parts of the y-axis as separate or not. I would say probably not, so the answer is three.

Answer 2

i think/know there are 2 asymptotes: x = 0 and y = x

Answer 3

x = 0, y = 0, y = +x

Answer 4

I'd say three - one vertical, one horizontal and one sloping.