Problem with basic definition of a tangent line.

I have just started studying calculus for the first time, and here I see something called a tangent. They say, a tangent is a line that cuts a curve at exactly one point. But there are a lot of lines that can cut the same point just like shown in the picture- enter image description here

WHY aren’t we saying that lines M and C are also tangents?

What is the real definition of a tangent line?

“ A tangent line is a line which passes through two infinitesimally close points” Is this definition correct?
Thanks!

Answer

Animations showing that the limit of a secant as the variable point tends towards a fixed point becomes the tangent line.

First, when the variable point (in red) approaches the fixed point (in black) from below:

enter image description here

Second, when the variable point approaches the fixed point from above:

enter image description here

In both cases, the secant line becomes the same tangent line. This suggests that the derivative is well-defined at the black point (which is highlighted in green when tangency occurs).

Formally, then, this gives us a definition of the derivative as follows. For a real-valued function f(x), the derivative at x = a is the limit of the slope of the secant line through two points P_a = (a, f(a)), \quad P_b = (b, f(b)) as b approaches a, whenever such a limit exists. Since the slope is simply \frac{f(b)-f(a)}{b-a}, we then have f'(a) = \lim_{b \to a} \frac{f(b) – f(a)}{b-a}. The equation of the tangent line at the point P_a is therefore y – f(a) = f'(a)(x-a), if the line does not have infinite slope; otherwise 1/f'(a) = 0 and we can write the equation of the tangent line as x = a.

Attribution
Source : Link , Question Author : Aaryan Dewan , Answer Author : heropup

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