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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 $

or $f(x)=x$

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?

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    $\begingroup$ Tangent line does not mean touching at one point. The tangent line is the best linear approximation to the function. $\endgroup$ – Ittay Weiss Aug 30 '17 at 22:59
  • $\begingroup$ It can be defined without using derivatives: see my answer below. I don't know of a way to define it without using limits, but that is by no means circular. A limit statement is nothing more than a quantified claim that one inequality implies another inequality. $\endgroup$ – G Tony Jacobs Aug 30 '17 at 23:21
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The line with equation $y=ax+b $ is tangent to the curve of equation $y=f (x) $ at the point $x=x_0$ if $x_0$ is a $\color {red}{double\; root } $ of the equation $$f (x)=ax+b $$

which means that

$$f (x_0)=ax_0+b$$ and $$f'(x_0)=a $$

or, in other words

$$f (x)-(ax+b)=(x-x_0)^2g (x)$$

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In calculus, we don't use the definition of "tangent" from elementary geometry. Instead, the definition goes like this: let $f(x)$ be a function, and $a$ a point in its domain. Then a line $g(x)$ is tangent to the graph of $f$ at $a$ if $g(a)=f(a)$ and the function $h(x)=f(x-a)-g(x-a)$ is smaller than any non-zero linear function as $x\to 0$, i.e., for every real $b>0$, there is an $\epsilon$ such that, for $0<|x|<\epsilon$, we have $|h(x)|<bx$

In less technical terms, if you zoom in on the graph at a point, and it starts to resemble a straight line, then the straight line that it starts to resemble is the tangent line.

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