Evaluate limits of a function

Question

Evaluate limx→3 x²−x−6/x²(x−3)²

If it does not exist because of infinite behavior, describe the infinite behavior using limit notation. 

Answer 1

lim_x→3 (x² - x - 6) / (x²)(x - 3)²

= lim_x→3 (x - 3)(x + 2)/ (x²)(x - 3)²

= lim_x→3 (x + 2)/ (x²)(x - 3)

d.n.e. (since when x = 3 numerator is ≠ 0 but denominator = 0)

lim_x→3- (x + 2)/ (x²)(x - 3) = - ∞ and lim_x→3+ (x + 2)/ (x²)(x - 3) = ∞

This rational function's graph has a vertical asymptote at x = 3. As x approaches 3 from the left, the function decreases without bound (heading toward - ∞). The function's y-values change signs at x = 3, so as x approaches 3 from the right, the function's y-values increase without bound (heading toward ∞).