Geometrical interpretation of derivative?

Statement: being unable to solve the famous derivative's limit at the point $P$ is actually equals to being unable to graph a tangent line to that point $P$.

Statement seems to be incomplete. At any given point one might graph as much tangent lines as he would like to. But the problem is that those tangents won't convey any "good" approximation for the original $f(x)$ at the given point $P$. As such, could I conclude something like:

Being unable to solve the famous derivative's limit at the point P is actually equals to being unable to graph a tangent line to that point P SUCH THAT IT PROVIDES "GOOD" LINEAR APPROXIMATION AROUND POINT P.

Or, slightly modified:

Among all possible tangent lines to the point P, there might be exactly one representing the best possible approximation; in case it exists, f(x) is differentiable at that point. Otherwise none of possible tangent lines to the point P gives "good" approximation.

Am I right?

P.S.

"Good" approximations means that $|f(x) - l(x)| \lt \epsilon$ for any $\epsilon \gt 0$ while $x \rightarrow P_x$.

• I think you misunderstand what "tangent line" means geometrically; a differentiable function can only have one tangent line at a given point. This is a geometric property, distinct from the related analytic properties. – Ian Dec 22 '17 at 19:09
• @Ian according to my knowledge, a tangent line is a line that shares at least 1 point with original function. Which means a tangent line to point $P$ of $f(x)$ shares... $P$ with it. Captain Obvious. – Sereja Bogolubov Dec 22 '17 at 19:10
• Nice functions like $f(x)=x^2$ are usually considered to have only one tangent at each point. So your definition of "tangent at $P$" as "any line that intersects the graph at $P$" runs contrary to the established definition. – Arthur Dec 22 '17 at 19:15
• @Arthur it probably has came historically due to obvious pointlessness of having infinite tangent lines which have no value, give no benefits. But hold a second. How many lines could you possible have passing single given point? So, being as restrict as possible, there is an infinity of them indeed. – Sereja Bogolubov Dec 22 '17 at 19:17
• So if I get this straight: you think the definition of "tangent" is bad, so you make your own? And then you come here because you can't make sense of results referring to the classical definition when you use your new definition? – Arthur Dec 22 '17 at 19:20