We know that a function $f(x)$ is continuous at a point $c$ if
$$\lim\limits_{x\to{c}}f(x)=f(c)$$ I am able to visualise it geometrically. What this means is that both the right hand and left hand limits at the point c are equal and are $f(c)$. In plain terms, that means if we go from the left hand side or the right hand side of the point c, the value of $f(x)$ is same and is equal to $f(c)$
A derivative is defined if the following limit exists:
$$\lim\limits_{h\to{0}}\frac{f(x+h)-f(x)}{h}$$ But what is the geometric interpretation of this limit not existing?
If this limit doesn't exist, then
$$\lim\limits_{h\to{0^+}}\frac{f(x+h)-f(x)}{h}≠\lim\limits_{h\to{0^-}}\frac{f(x+h)-f(x)}{h}$$
But I am not able to visualise what this geometrically means?
It would help me immensely if I would be able to visualise it on a graph.
Question:
How to visualise it geometrically (or graphically) if this limit does not exists?$$\lim\limits_{h\to{0}}\frac{f(x+h)-f(x)}{h}$$
The derivative is the limit of the slope of the secant, if the right hand and left hand derivative exist but are not equal then that means the limit of slope of the secant from end does not equal the limit of the slope of the secant from the other end.