NOTE: Due to formatting constraints, limits are shown with a horizontal bar in these quiz questions.For example: will be written as: [math: limx→af(x)] \frac{lim}{x \rightarrow a }f(x) Solve the following limit and write your answer to three decimal places.[math: limx→2(x3+7x2−18xx2−4)] \frac{lim}{x \rightarrow 2 } (\frac{x^3+7x^2-18x}{x^2-4})

Question

BlackTom AI · Accepted Answer

The problem asks us to evaluate a limit of a rational expression as x approaches 2.
First, I look at the numerator: x^3 + 7x^2 − 18x. This can be factored by pulling out an x: x(x^2 + 7x − 18).
Then I factor the quadratic: x^2 + 7x − 18 = (x + 9)(x − 2). So the numerator becomes x(x + 9)(x − 2).
Next, the denominator x^2 − 4 is a difference of squares and factors as (x − 2)(x + 2).
We now have the expression (x(x + 9)(x − 2)) / ((x − 2)(x + 2)). Since x ≠ 2 near the limit, we can cancel the common factor (x − 2).
This leaves the simplified expression x(x + 9)/(x + 2).
Now substitute x = 2 into the simplified expression: 2(2 + 9)/(2 + 2) = 2 · 11 / 4 = 22/4 = 5.5.
Therefore, the limit exists and equals 5.5, which to three decimal places is 5.500.
Note: If you were to skip factoring, you might incorrectly evaluate a non-existent direct substitution; factoring and canceling the common factor (x − 2) is essential to resolving the removable discontinuity at x = 2.

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