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 (g(x)-g(-3))/(x+3)

The limit has a value of -1/3

Step-by-step explanation:

The derivative of a curve at a point represents the slope of the tangent line at a point. Thus if we can obtain the slope of the given line equation that is tangent to the graph g(x), we will know the value of the limit.

Since the limit of the difference quotient represents the derivative which is the slope of the tangent line.

Identifying the slope of the tangent line

We can proceed solving for y from the given equation

Subtracting by 3 both sides and distributing -2/3 to each term inside the parenthesis give us

Then we can divide both sides by 2

We have the equation in slope-intercept form, y = mx+b, so we can tell that the slope is -1/3, so we can write

Thus the limit of the difference quotient is -1/3 as well since it represents the slope of the tangent line.

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