Question

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)?

Answer 1

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