Sorry if my English is wrong.
In calculus, given a function $f$, derivable at $x_0$, the tangent line to the curve at $x_0$ is
$$t(x) = f(x_0) + f'(x_0)(x-x_0)$$
How can I convince myself that this line corresponds to the intuition of "a line is tangent to a curve if it intersects the curve one and only one time". I think you can get a segment of line that verifies this for all non-linear functions.
The reason I say segment of line, and not line is because, for example if $f(x)=x^3$, the tangent line to the graph of $f$ at $x_0=-0.5$, $t(x)=\frac{\left(3x\ +\ 1\right)}{4}$intersects the curve at $x=-0.5$ and $x=1$.
How to prove that $t$ is such line, and no other? Thank you.
That's not actually the definition of the tangent line, because for example the line $x = 0$ touches $y = x^3$ at only one point, but is not a tangent line.
A better version might be "the slope of the line passing through two points of the curve, in the limit as the two points approach each other and become one." It's a much more specific criterion.