Complete question 1, 4 and 8 only.
** Please see the attached file for the complete solution response **
(Q.1) It is said that a function (please see the attached file) is differentiable at a point a (please see the attached file) R if:
a) f is continuous at the point a
b) (please see the attached file) the limit
does exist. In such a case, this limit is called "the derivative of f(x)" at the point a and it is denoted as f' (a).
Remark: The condition of continuity is absolutely necessary, because it is possible the limit (1) to exist, but the function not to be continuous. Example:
Since f(x) is a constant on both sides of x = 0, we have :
(please see the attached file)
but f is not continuous at x = 0, therefore it is not differentiable at x = 0.
It is said that a function has a limit at the ...
This solution provides a detailed, step-by-step explanation of the given calculus problems.