The line defined by the equation 2y+3 = - (2/3) (x-3) is tangent to the graph of g (x) at x=-3. What is the value of the limit as x approaches - 3 of [g (x) - g (-3) ]over (x+3) ?

The limit as x approaches - 3 of [g (x) - g (-3) ] / [x - (-3) ] is, by definition of limit, g ' (x) at x = - 3.

And g ' (x) at x = - 3 is the slope of the line tangent to it.

2y + 3 = - (2/3) (x-3)

2y = - 2x/3 + 2 - 3

y = - x/3 - 1/2 = > slope = - 1/3

Then the limit is - 1/3

Answer: - 1/3