Quod Erat Demonstrandum

2007/11/02

[AL][PM] Asymptote

Filed under: HKALE,Pure Mathematics — johnmayhk @ 12:05 下午

A livejournal writer, Mr. Ng, felt surprising that many Hong Kong secondary school mathematics teachers have no idea in the definition of asymptote. He quoted that “$\lim_{x\rightarrow \infty} f(x) = 0 \Rightarrow x$-axis is a horizontal asymptote of $y = f(x)$" is completely wrong! He then gave a counter-example $y = \frac{\sin(x)}{x}$. He says that even in MathWorld, the definition that “An asymptote is a line or curve that approaches a given curve arbitrarily closely." is 一塌糊塗。Some times ago, I found the following in the internet.

From a page of an article “The Asymptotes of Plane Curves" written by H. G. Green, published in “The Mathematical Gazette" (Vol. 13, No. 185 (Dec., 1926), pp. 232-235), we see a definition of asymptote as follows.

An asymptote of a curve of the $n^{th}$ degree is the limit of a continuous series of parallel lines which cut the curve in less than $n$ points and whose intersections therewith, as the lines approach the limit, become and remain farther from the origin than any given distance, however great."

As in Mr. Ng’s further discussion, there are MANY definitions for asymptote. I think it is a good topic to study further.

3 則迴響 »

1. >An asymptote of a curve of the n^{th} degree is the limit of a continuous series of parallel lines which cut the curve in less than n points and whose intersections therewith, as the lines approach the limit, become and remain farther from the origin than any given distance, however great.”

Fancy. It is quite nicely constructed as far as I can see (i.e. being confirmed with as many examples as my brain could enumerate), but with such complexity we can’t easily investigate its correspondence to our intuitive understanding of asymptotes… it reminds me with this thing about the widely accepted definition on continuity:

Given a function f: Real numbers -> Real numbers on an interval I, and a point p in I,
For all E>0, there exists d>0 such that

x belongs to I and | x – p | f(x) Real numbers

f(x) = 0, for all irrational x.

f(x) = 1/(p+q), with any rational x, having p/q as the simplest quotient form (i.e. HCF(p,q) = 1)

It is an interesting result that f is continuous at any irrational number, and discontinuous only at rational numbers, in the given interval. It is virtually impossible to confirm that the definition would be the same as what we have in our mind, within the scope of this particular case.

迴響 由 Ei — 2008/05/27 @ 8:14 上午 | 回應

2. Oops, something as gone wrong with my text. Please ignore the previous comment entry.

>An asymptote of a curve of the n^{th} degree is the limit of a continuous series of parallel lines which cut the curve in less than n points and whose intersections therewith, as the lines approach the limit, become and remain farther from the origin than any given distance, however great.”

Fancy. It is quite nicely constructed as far as I can see (i.e. being confirmed with as many examples as my brain could enumerate), but with such complexity we can’t easily investigate its correspondence to our intuitive understanding of asymptotes… it reminds me with this problem assuming the widely accepted definition of continuity:

We define f: (0,1] -> Real numbers

f(x) = 0, for all irrational x.

f(x) = 1/(p+q), with any rational x, having p/q as the simplest quotient form (i.e. HCF(p,q) = 1)

It is an interesting result that f is continuous at any irrational number, and discontinuous only at rational numbers, in the given interval. It is very hard, to confirm intuitively that, the given definition is the same as what we have in our minds, within the scope of this particular case.

迴響 由 Ei — 2008/05/27 @ 8:22 上午 | 回應

3. *has gone wrong

>f(x) = 1/(p+q), with any rational x, having p/q as the simplest quotient form (i.e. HCF(p,q) = 1)

as the simplest quotient form of x, i.e. HCF(p,q) = 1 & x=p/q

Sorry to have made so many mistakes… =.=

迴響 由 Ei — 2008/05/27 @ 8:24 上午 | 回應