How to solve this math by k(x+h)-k(x)/h this formula?

Question

Consider the piecewise function

g(x)={(1+x)⁴−1, if x≥0

sin(4x), if x<0.

Discover whether g(x) is differentiable at x=0 using the limit definition of the derivative. Without carrying out any further calculation, what can you conclude about the continuity at point x=0?

Answer 1

1st you need to decide if the function is continuous at x=0 (if a function is not continuous at x=0, then it can't be differentiable at x=0)

How do you decide if a piecewise function is continuous at x=0?

If f(x) is continuous at x=0, then we can look into whether or not f(x) is differentiable at x=0.

The derivative is a limit, so you need to confirm that the left hand and right hand limits of the quotient function are equal to one another. If they are, then the function is differentiable at x=0.