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I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me if he is right.

Definition:

Suppose two curves $C_1$ and $C_2$ have a regular point P in common. Take a point A on $C_1$ near P and let $D_A$ be the point on $C_2$ closest to A (i.e orthogonal projection of A on $C_2$) then $C_2$ has contact of order $n$ with $C_1$ at P if

$ \lim_{A\to P} \dfrac{ \mbox{dist}(A,D_A)}{[\mbox{dist}(A,P)]^k}= \begin{cases} c\neq 0, & \text{if } \; k=n+1 \\ 0, & \text{if } \; k\leq n \end{cases}$


Argument: $\lim_{A\to P} \frac{AD_A}{(AP)^{n+1}} =c \neq 0$ then $\lim_{A\to P} \frac{AD_A}{(AP)^{n}} = \lim_{A\to P} AP \cdot \frac{AD_A}{(AP)^{n+1}}$

$ = \lim_{A\to P} AP \cdot \lim_{A\to P} \frac{AD_A}{(AP)^{n+1}} = 0 \cdot c =0$

The same process can be followed to show that the limit is zero for all $k\leq n$

Note: The above definition is claimed by the authors to be extracted from this original source

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    @Srivatsan if it is not finite then we cannot conclude that $0\cdot c =0$ and the argument won't stand.2011-09-16

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