Question

Is the tangent line a point on a line always the line itself?

Answer 1

An unrelated point is an infinite-directional null vector.. Yet, as point of contact of a rotating tangent with the curve, its direction is fixed.

Explanation:

Despite that the question is not clear, it seeks some truth about points. Perhaps, the basis for the question is that the direction of the tangent is the limiting direction, when two neighboring points on the curve merge to the point of contact, in the limit.

This is my answer from my understanding of the question.

A null vector can be regarded as the limit of a vector to a point, when the modulus of the vector tending to 0. The direction of this null vector is the direction in which the parent vector tends to the null vector.

Independently, a point is a null vector that is infinite-directional, and is available for association with any direction, due to the traversing of a line/curve through the point.

For a tangent, the point of contact is a null vector, in the direction of the tangent.

Answer 2

Here's my guess as to what the question was supposed to be. "Is the line tangent at a point on a line always the line itself?"

Explanation:

Yes.

For a non-vertical line `f(x) = y=mx+b`, if we fix a point `(a,f(a))`, then

`lim_(hrarr0)(f(a+h)-f(a))/h = lim_(hrarra) ([m(a+h)+b]-[ma+b])/h`

`= lim_(hrarr0)(mh)/h = m`

The line through `(a, f(a))` with slope `m` is exactly the line we started with.

For a vertical line, we would need a definition of the line tangent to a vertical line. I've seen presentations of calculus that define "vertical tangent line at `(a,f(a))` by using `lim_(xrarra)abs(f'(x)) = oo`. That won't work if we start with a vertical line, because

1. it isn't the graph of a function, and

2. We can't talk about what happens to the secant lines as `x` approaches a fixed value. The problem is that, even after we make sense out of secant lines to a vertical line, all secant lines are vertical.

On the other hand. Given the result above for non-vertical lines, it is reasonable to define the tangent line to a vertical line to be the vertical line itself.