: limit(x-3^-)f'(x)=−∞and limit (x-3^+)f'(x)=∞

What is the meaning of : limt(x-3^-)f'(x)=−∞and limit (x-3^+)f'(x)=∞ ????

This means that the rate of change, or the slope of f(x) approaches negative infinity as x approaches 3 from the negative side. I.E., it approaches 3 coming from 0 to 1 to 2 to .. as close to 3 as possible. In reverse, the next limit means that the derivative is approaching positive infinity from the positive side of 3.
With that being said, what could this mean for f(x)? It means that f(x) is undefined at x = 3. The reason being that the derivative is not defined for the limit as x approaches 3 from both sides at the same time. This means the function is discontinuous at x = 3.