I was unsure about what exactly the definition of a tangent line is given that in the traditional sense, it is a line that "touches" a function at a single point. However, certain functions have derivatives that cross the function at more than two points such as
$ f(x) = sin(x) $
at $f'(0) = 0 $
where the derivative will be $f'(x)=1$
both of which intersect the function at multiple points. Furthermore, can we define a tangent line without using derivatives and limits (because doing so seems to make things little circular) or are derivatives absolutely necessary to define a tangent line?