I found a similar question and a beautiful answer here. However I'm not able to fully understand the answer and have a question on the selected answer at:
Consider all the lines going through point $(x_0,f(x_0))$. For every line, the relative error should approach $0$ as $x$ approaches $0$ because all these lines go through point $(x_0,f(x_0))$, the linear approximation equals the function at $x=x_0$. What is special about the tangent line in relation to the relative error? Why does Arturo say only the tangent line makes the relative error zero ?