Question

Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value. x? lim -9 X-3 X+3 Simplify the rational expression. x²-9 x+3 Evaluate the limit or determine that it does not exist. Select the correct choice below and, if necessary, fill in the answer box within your choice. ОА, 9 lim X-3X+3 (Simplify your answer.) B. The limit does not exist and is neither co nor - 00.

Answer 1

Answer: B.
Given expression: (-9x) / (x^2 - 9)
Simplify the rational expression by factoring the denominator:
(x^2 - 9) = (x + 3)(x - 3)
= (-9x) / [(x + 3)(x - 3)]

Now, we can evaluate the limit as x approaches 3:
lim (x -> 3) [(-9x) / ((x + 3)(x - 3))]

Since the expression is defined and continuous at x = 3, we can directly substitute the value of x:
((-9 * 3) / ((3 + 3)(3 - 3))) = (-27) / (6 * 0)

The denominator becomes zero, which means the limit does not exist, and is neither ∞ nor -∞. So, the correct choice is B. The limit does not exist and is neither ∞ nor -∞.