Question

Let f(x) be a function defined for all real numbers, such that f(3x+9) = f(x) for all x in mathbb{R}. If lim_(x→-2) [f(x) - 3x^2 + 7] / (x-2) = 12, find (if it exists).

Answer 1

The limit does not exist for the given function and expression.

Step-by-step explanation:

To find the value of the limit lim(x→-2) [f(x) - 3x^2 + 7] / (x-2) = 12, we can use the property of continuity and the given function equation f(3x+9) = f(x). Since the limit is taken as x approaches -2, we can substitute x = -2 into the equation f(3x+9) = f(x) to get f(-6) = f(-2).

With this information, we can rewrite the original limit expression as: lim(x→-2) [f(x) - 3x^2 + 7] / (x-2) = lim(x→-2) [(f(x) - f(-6)) - 3(x^2 - 4)] / (x-2).

Since we know that lim(x→-2) (x-2) is equal to 0, we can simplify the expression further:

lim(x→-2) [(f(x) - f(-6)) - 3(x^2 - 4)] / (x-2) = lim(x→-2) [(f(x) - f(-6)) - 3(x+2)(x-2)] / (x-2).

By canceling out the common factor of (x-2), we get:

lim(x→-2) (f(x) - f(-6)) - 3(x+2) = 12.

Since f(-6) = f(-2), we can simplify the equation further:

lim(x→-2) (0) - 3(-2+2) = 12.
0 - 3*0 = 12.
0 = 12.

Since the equation 0 = 12 is not true, it means that the limit does not exist for the given function and expression.