Does the tangent line really touch a single point?

Question

The principle of the theory of the tangent to a curve is that it is the limit of the secant when the two intersection points, one of them approaches the other, I mean in Calculus I. I intend to prove that for every function $f$ differentiable on an interval $I$ and a number $a$ from that interval, the equation : $f’(a)(x-a)+f(a)=f(x)$ accepts $a$ as a double solution. Can I? Or prove that their intersection is a double point.

Answer 1

If $f$ is differentiable on an interval $(b, c)$, then for any $b \lt a \lt c$ such that $f''(a)$ exists, we have, after an application of L'Hôpital's rule, $$\lim\limits_{x \rightarrow a} \frac{f'(a)(x - a) + f(a) - f(x)}{(x - a)^2} = \lim\limits_{x \rightarrow a} \frac{f'(a) - f'(x)}{2(x - a)} = -\frac{1}{2}f''(a).$$

Thus $a$ is a zero of order larger than or equal to $2$ in this case (note that it could even be larger than $2$ if the $f''(a) = 0$).

The counterexamples given in the comments above have functions that are not twice-differentiable at the point where the limit is taken so this argument does not work for them. One could also come up with functions where the solution $a$ is a $n$-th order zero for an arbitrary $n \gt 1$.

For example $f(x) = x^5$ has the equation $f'(0)(x - 0) + f(0) - f(x) = -x^5 = 0$ with $0$ as a solution with multiplicity $5$.