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!
Best 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:
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 $x = a$.