这样的直线是否一定是切线呢？

manthanein

在坐标平面上给定曲线\(x=\phi(t), y=\psi(t)\)。\(t\)的定义域另外指定为\(D\)。\(\phi(t)\)和\(\psi(t)\)在定义域\(D\)上都连续。
取\(t=a \in D\)，\(a\)为常数。构造直线\(L\)，\(A\)、\(B\)均为常数：
\(L：A[x-\phi(a)]+B[y-\psi(a)]=0\)
取\(u \in D\)，构造：
\(F(u)=[\phi(u)-\phi(a)]^2+[\psi(u)-\psi(a)]^2\)
\(G(u)=\D \frac{\D \abs{A[\phi(u)-\phi(a)]+B[\psi(u)-\psi(a)]}}{\sqrt{A^2+B^2}\cdot F(u)}\)
已知：
对于任意给定的正数\(\epsilon\)，总存在正数\(\delta\)，使得\(F(u) \lt \delta\)时，一定有\(G(u) \lt \epsilon\)。
求可能的直线\(L\)。
从几何角度看，直线\(L\)应该是\(t=a\)处的切线。
但是几何直观未必正确，能否有一个严格的代数证明呢？
当然，也有可能不是切线，如果不是切线，又会是什么？

补充内容 (2016-12-1 22:52):
\(F(u)=\sqrt{[\phi(u)-\phi(a)]^2+[\psi(u)-\psi(a)]^2}\)更正

补充内容 (2016-12-1 22:53):
\(D\)为一个区间

补充内容 (2016-12-2 21:52):
\(u\)不在给定直线上

manthanein

先试试吧。
先假设：\(\phi(u) \ne \phi(a)\)，\(\psi(u) \ne \psi(a)\)，\(B \ne0\)，\(a\)不在定义域边界上
令\(k=\D -\frac{A}{B}\)
\(G(u)\)
\(=\D \frac{\D \abs{A[\phi(u)-\phi(a)]+B[\psi(u)-\psi(a)]}}{\sqrt{A^2+B^2}\cdot F(u)}\)
\(=\D \frac{\D \abs{-Bk[\phi(u)-\phi(a)]+B[\psi(u)-\psi(a)]}}{\sqrt{B^{2}k^{2}+B^2}\cdot F(u)}\)
\(=\D \frac{\D \abs{B}\abs{-k[\phi(u)-\phi(a)]+[\psi(u)-\psi(a)]}}{\abs{B} \sqrt{k^{2}+1}\cdot F(u)}\)
\(=\D \frac{\D \abs{-k[\phi(u)-\phi(a)]+[\psi(u)-\psi(a)]}}{\sqrt{k^{2}+1}\cdot F(u)}\)

manthanein

微积分理论告诉我们，此时存在切线的充分必要条件是存在常数\(C\)满足：
\(\D \lim_{u \to a} \D \frac{\psi(u)-\psi(a)}{\phi(u)-\phi(a)}=C\)

manthanein

也就是说：
对于任意给定的正数\(\lambda_1\)，存在正数\(\gamma_1\)
使得只要\(u\)满足不等式\(0 \lt \abs{u-a} \lt \gamma_1\)，一定有\(\abs{\D \frac{\psi(u)-\psi(a)}{\phi(u)-\phi(a)}-C} \lt \lambda_1\)

manthanein

令\(k=C+\zeta\)
\(G(u)\)
\(=\D \frac{\D \abs{-(C+\zeta)[\phi(u)-\phi(a)]+[\psi(u)-\psi(a)]}}{\sqrt{k^{2}+1}\cdot F(u)}\)
\(= \D \frac{\D \abs{[\psi(u)-\psi(a)]-(C+\zeta)[\phi(u)-\phi(a)]}}{\sqrt{k^{2}+1}\cdot F(u)}\)
\(= \D \frac{\D \abs{[\psi(u)-\psi(a)]-C{[\phi(u)-\phi(a)]}-\zeta[\phi(u)-\phi(a)]}}{\sqrt{k^{2}+1}\cdot F(u)}\)
\(\le \D \frac{\D \abs{[\psi(u)-\psi(a)]-C{[\phi(u)-\phi(a)]}}+\abs{\zeta[\phi(u)-\phi(a)]}}{\sqrt{k^{2}+1}\cdot F(u)}\)
\(= \D \frac{\D \abs{[\psi(u)-\psi(a)]-C{[\phi(u)-\phi(a)]}}+\abs{\zeta}\abs{[\phi(u)-\phi(a)]}}{\sqrt{k^{2}+1}\cdot F(u)}\)

manthanein

另一方面：
\(G(u) \ge \D \frac{\D \abs{[\psi(u)-\psi(a)]-C{[\phi(u)-\phi(a)]}}-\abs{\zeta}\abs{[\phi(u)-\phi(a)]}}{\sqrt{k^{2}+1}\cdot F(u)}\)

manthanein

\(\abs{\D \frac{\psi(u)-\psi(a)}{\phi(u)-\phi(a)}-C} \lt \lambda_1\)
\(\D \frac{\abs{[\psi(u)-\psi(a)]-C{[\phi(u)-\phi(a)]}}}{\abs{\phi(u)-\phi(a)}} \lt \lambda_1\)
\(\abs{[\psi(u)-\psi(a)]-C{[\phi(u)-\phi(a)]}} \lt \lambda_1{\abs{\phi(u)-\phi(a)}}\)

manthanein

\(G(u) \le \D \frac{\D \abs{[\psi(u)-\psi(a)]-C{[\phi(u)-\phi(a)]}}+\abs{\zeta}\abs{[\phi(u)-\phi(a)]}}{\sqrt{k^{2}+1}\cdot F(u)} \lt \frac{\lambda_1{\abs{\phi(u)-\phi(a)}}+\abs{\zeta}\abs{[\phi(u)-\phi(a)]}}{\sqrt{k^{2}+1}\cdot F(u)} \)

manthanein

\(\D \frac{\lambda_1{\abs{\phi(u)-\phi(a)}}+\abs{\zeta}\abs{[\phi(u)-\phi(a)]}}{\sqrt{k^{2}+1}\cdot F(u)}\)
\(=\D \frac{\lambda_1{\abs{\phi(u)-\phi(a)}}+\abs{\zeta}\abs{[\phi(u)-\phi(a)]}}{\sqrt{k^{2}+1}\cdot \sqrt{[\phi(u)-\phi(a)]^2+[\psi(u)-\psi(a)]^2}}\)
\(=\D \frac{\lambda_1{\abs{\phi(u)-\phi(a)}}}{\sqrt{k^{2}+1}\cdot \sqrt{[\phi(u)-\phi(a)]^2+[\psi(u)-\psi(a)]^2}}+\frac{\abs{\zeta}\abs{[\phi(u)-\phi(a)]}}{\sqrt{k^{2}+1}\cdot \sqrt{[\phi(u)-\phi(a)]^2+[\psi(u)-\psi(a)]^2}}\)

manthanein

\(\D \frac{\lambda_1{\abs{\phi(u)-\phi(a)}}}{\sqrt{k^{2}+1}\cdot \sqrt{[\phi(u)-\phi(a)]^2+[\psi(u)-\psi(a)]^2}}+\frac{\abs{\zeta}\abs{[\phi(u)-\phi(a)]}}{\sqrt{k^{2}+1}\cdot \sqrt{[\phi(u)-\phi(a)]^2+[\psi(u)-\psi(a)]^2}}\)
\(= \D \frac{\lambda_1}{\sqrt{k^{2}+1}\cdot \sqrt{1+[\D \frac{\psi(u)-\psi(a)}{\phi(u)-\phi(a)}]^2}}+\frac{\abs{\zeta}}{\sqrt{k^{2}+1}\cdot \sqrt{1+[\D \frac{\psi(u)-\psi(a)}{\phi(u)-\phi(a)}]^2}}\)