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- 

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!
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:



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



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 $x = a$.
The tangent line is the unique line that "touches" the point without actually actually crossing it. For the other lines in your example, there's a crossover at the point; the line goes from being above the curve to below it. The tangent line remains beneath it.

This is obviously not the rigorous definition, but for someone just starting out with calculus this is probably the easiest way to conceptualise it.