AP Calculus AB Exam Question

Consider the function defined by:

f(x) = (x^3 - 4x^2 - 3x + 18)/(x^2 - 2x - 3)

Determine the limit of f(x) as x approaches 3 algebraically.

Solution

To find the limit of f(x) as x approaches 3, we first factorize the polynomial in the numerator and denominator:

f(x) = ((x - 3)(x^2 + 2x - 6))/((x - 3)(x + 1))
     = (x^2 + 2x - 6)/(x + 1)

Now, we can cancel out the common factors of (x - 3):

f(x) = (x^2 + 2x - 6)/(x + 1)
     = (x - 1)(x + 3)/(x + 1)

Since (x - 1) and (x + 3) are both continuous functions, we can evaluate the limit at x = 3 by plugging in the value:

f(3) = (3 - 1)(3 + 3)/(3 + 1)
     = 2(6)/4
     = 12/4
     = 3

Therefore, the limit of f(x) as x approaches 3 is 3.

Answer: The limit of f(x) as x approaches 3 is equal to 3, i.e., lim(x->3) f(x) = 3.