Evaluate lim_(x->0) ((x+b)-b)/x

Step 1: Evaluate the limit for Question 16. First, simplify the numerator of the expression. _x 0 ((x+b)-b)/(x) = _x 0 (x)/(x) Since x 0, x is not exactly zero, so we can cancel x from the numerator and denominator. _x 0 1 = 1 Answer for 16: 1 Step 2: Evaluate the limit for Question 17. If we substitute x=0 directly, we get (0)/(0), which is an indeterminate form. We need to simplify the expression. Expand the numerator: (x+3)^2 - 9 = (x^2 + 6x + 9) - 9 = x^2 + 6x Now substitute this back into the limit expression: _x 0 (x^2 + 6x)/(x) Factor out x from the numerator: _x 0 (x(x+6))/(x) Since x 0, x is not exactly zero, so we can cancel x: _x 0 (x+6) Now substitute x=0: 0+6 = 6 Answer for 17: 6 Step 3: Evaluate the limit for Question 18. If we substitute h=0 directly, we get (0)/(0), which is an indeterminate form. We need to simplify the expression. Expand the numerator (x+h)^3: (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 Now substitute this back into the numerator of the limit expression: (x^3 + 3x^2h + 3xh^2 + h^3) - x^3 = 3x^2h + 3xh^2 + h^3 Substitute this simplified numerator back into the limit: _h 0 (3x^2h + 3xh^2 + h^3)/(h) Factor out h from the numerator: _h 0 (h(3x^2 + 3xh + h^2))/(h) Since h 0, h is not exactly zero, so we can cancel h: _h 0 (3x^2 + 3xh + h^2) Now substitute h=0: 3x^2 + 3x(0) + (0)^2 = 3x^2 + 0 + 0 = 3x^2 Answer for 18: 3x^2 3 done, 2 left today. You're making progress.