Fermat's method of calculus

I've attached my question as a file and here is the resource I was looking at https://cedar.wwu.edu/cgi/viewcontent.cgi?article=1012&context=wwu_honors

  • what is the question?

  • you are asking why does this actually work even after answering the same. But you are also sayng you wont get anywhere

  • Your question is very vague. Please clarify.

  • my question basically is why does this method work to give you a tangent line? specifically the steps where you divide by e and then plug 0 in for e

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  • can you explain further the last part you wrote?

  • In essence, this procedure allows to find the x-axis cross value as a limit operation. When we replace T(x+epsilon) by f(x+epsilon) we are thinking of a sufficiently small epsilon for this to make sense. This is a limit operation t that is quite literally based on Taylor first order approximation.

  • I added another explanation.

The answer is accepted.