# Quod Erat Demonstrandum

## 2014/05/21

### 與漸近線相交

Filed under: Fun,Pure Mathematics — johnmayhk @ 1:35 下午
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「人工」地使圖像產生「劇烈」變化，比如用絕對號，見 2009 HKAL Pure Mathematics Paper II Q.7，考慮 $y=\frac{|x|(x+16)}{x-2}$ 如下：

$\left \{ \begin{array}{ll} x=t+\frac{\cos(nt)}{t} &\\ y=t+\frac{\sin(nt)}{t} &\end{array}\right.$

$x\rightarrow \pm\infty \Rightarrow t\rightarrow \pm\infty$

$\displaystyle \lim_{x\rightarrow \pm\infty}\frac{y}{x}=\displaystyle \lim_{t\rightarrow \pm\infty}\frac{t+\frac{\sin(nt)}{t}}{t+\frac{\cos(nt)}{t}}=1$

$\displaystyle \lim_{x\rightarrow \pm\infty}(y-x)=0$

$\left \{ \begin{array}{ll} x=t+\frac{\cos(nt)}{t} &\\ y=t+\frac{\sin(nt)}{t} &\end{array}\right.$

$n=4$

$n=8$

$n=12$

$n=16$

$\left \{ \begin{array}{ll} x=t+\frac{\cos(5t)}{t} &\\ y=t+\frac{\sin(10t)}{t} &\end{array}\right.$

（圖片來源：Kate Foster）

## 1 則迴響 »

1. A much simpler example is
f(x)=(sin x)/x

Clearly, y=0 is a horizontal asymptote and f(x) has infinitely many zeroes.

迴響 由 lovely girl — 2014/08/29 @ 1:15 上午 | 回應