# asymptotic infinity

I am very very new to math as a whole, so please excuse my n00biness.

I read: An asymptote of a curve is a line that continually approaches the curve but never meets it at any finite distance. The distance between the line and the curve approaches zero as they tend to infinity. When we say variable n tends to infinity it means as n gets very very large. If we look at 1/n as n tend to infinity, then n gets very large and 1/n goes to zero

So my question is: Based on the above can 1/n be analogous to the distance between a line and a curve, and n thought of as the increase in the size of the two?

• Do you have any specific kind of curve and line in mind? ;) Commented May 24, 2013 at 15:29
• @Joachim: I am not even sure what I am to imagine... The thought in the question came to my mind, and I wanted to know if it is totally and unspeakably stupid.. Commented May 24, 2013 at 15:33
• There are a lot of curves imaginable that satisfy your needs, but there is one choice which would come first to mind for most people. It is given in the answers below. Commented May 24, 2013 at 15:36
• "But never meets it at any finite distance" practically no modern authors include this part! Commented May 24, 2013 at 15:37
• @WillieWong: Is that good or bad thing? Commented May 24, 2013 at 15:42

If you look at the picture:

As you plug larger and larger values for $x$, the blue curve will get closer and closer to the horizontal black line. The distance between the blue curve and black line is given by $\frac{1}{x}$, which gets smaller the bigger $x$ gets. $x$ in this case represents how far to the right we go -- the further to the right, the closer the two are.

• wow...that was such a wonderful answer! You, sir, are awesome! Commented May 24, 2013 at 15:36
• Nice one! We were writing at the same time i guess.. =) Commented May 24, 2013 at 15:37
• I wish I could give both of you more than one +1s...thanks for helping my interest in Math. Commented May 24, 2013 at 15:41
• Notice also that as $x$ becomes very very small...then approaching the line $x = 0$ from the right, along the curve, brings us very close to the y axis, heading upward. Approaching the line $x = 0$ from the left, along the curve, brings us very very close to the y-axis, heading downward: As $x \rightarrow 0$ from the right, $y \to +\infty$, as $x \rightarrow 0$ from the left, $y \to -\infty$! Commented May 24, 2013 at 15:44

In math, $n$ is normally used for an integer number, so let me switch to the variable $x$.

Consider the $(x,y)$-plane. Write $L$ for the set of points whose $y$ coordinate is zero and $C$ for the graph of the function $x \mapsto 1/x$. In other words: \begin{align*} L &= \{ (x,0) \quad | \quad x \in \mathbb{R} \}\\ C &= \{ (x,1/x) \quad | \quad x \in \mathbb{R} \} \end{align*} Note that $L$ is also a graph but a really stupid one, it corresponds to the function $x \mapsto 0$.

We run into trouble at $x=0$, so let us just consider the part of the plane where $x>0$.

If you sketch both $L$ and $C$ you will see that at some fixed $x$, the distance between them is $1/x$. Since $1/x$ gets smaller and smaller as $x$ increases, indeed their distance tends to zero. So yes, your $1/n$ can be thought of in this way (and $L$ is an asymptote for $C$).

• thanks for the answer, it helped me :) Commented May 24, 2013 at 15:38