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