Final answer:
The correct limit definition for g'(2) is given by using the difference quotient formula and simplifying it to find the value of the derivative as h approaches 0. Hence, the first option is correct.
Step-by-step explanation:
The correct limit definition of g'(2) for the function g(x) = -x² + 3x is determined by using the difference quotient formula. The definition of a derivative at a point x=a is the limit as h approaches 0 of the function (g(a+h) - g(a))/h. Therefore, for calculating g'(2), the correct expression using the limit definition is:
lim(h →0) [g(2+h) - g(2)] / h
where g(2+h) = -(2+h)² + 3(2+h) and g(2) = -2² + 3(2). So, the expanded form will be lim(h →0) [(-2(2+h) + 3(2+h)) - (-4+6)] / h, which simplifies down to:
lim(h →0) [(-4 - 2h + 6 + 3h) - (2)] / h
lim(h →0) [(-4 - 2h + 6 + 3h + 4 - 6) / h]
lim(h →0) [h] / h = lim(h →0) 1
Thus, the first option is correct: 1) lim(h →0) [(-2(2+h) + 3(2+h) - (-2(2)) + 3(2)) / h].