To evaluate or approximate limits after finding a difference quotient, algebraic manipulation is often necessary, particularly to cancel out the Δx in the denominator as it approaches zero. The process involves several steps: first, find f(x+h) by substituting x with x+h, then subtract f(x) and simplify. After that, divide by h and factor to reduce the expression. Finally, take the limit as h approaches zero to find the derivative. The example provided demonstrates that the derivative of f(x) = 4 - 2x - x² is f'(x) = -2 - 2x.