NOTE: Due to formatting constraints, limits are shown with a horizontal bar in these quiz questions.For example: will be written as: $ \frac{lim}{x \rightarrow a }f(x) $Solve the following limit and write your answer to three decimal places.$ \frac{lim}{x \rightarrow 2 } (\frac{x^3+7x^2-18x}{x^2-4}) $

Question

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Question: Solve the following limit and write your answer to three decimal places.

We are asked to find the limit as x approaches 2 of the expression (x^3 + 7x^2 - 18x) / (x^2 - 4).

Step 1: Factor both numerator and denominator if possible. The numerator x^3 + 7x^2 - 18x has a common factor of x, giving x(x^2 + 7x - 18). Now factor the quadratic: x^2 + 7x - 18 = (x + 9)(x - 2). Thus the numerator factors to x(x + 9)(x - 2).
Step 2: Factor the denominator: x^2 - 4 is a difference of squares, so (x^2 - 4) = (x - 2)(x + 2).
Step 3: Cancel common factors. The expression becomes [x(x + 9) (x - 2)] / [(x - 2)(x + 2)]. For x ≠ 2, (x - 2) cancels, leaving x(x + 9) / (x + 2).
Step 4: Evaluate the limit by direct substitution since the cancellation removed the problematic factor. Substitute x = 2 into the simplified expression: 2(2 + 9) / (2 + 2) = 2(11) / 4 = 22/4 = 5.5.

Answer: 5.5

UCIC 202503 EMTH118 Quiz 6

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