Question: "Evaluate the limit. limx→3x3−27x2−9"

High School
Algebra
Algebra I
Quadratic function and equation

Globokim8

Answered question

2021-01-28

Evaluate the limit.
$lim_{x \to 3} \frac{x^{3} - 27}{x^{2} - 9}$

Answer & Explanation

Nola Robson

Skilled2021-01-29Added 94 answers

For these sorts of problems, you always need to cancel something from the numerator and denominator before the limit can be evaluated. In this case, notice that

x^{2} - 9 = (x - 3) (x + 3) and x^{3} - 27 = (x - 3) (x^{2} + 3 x + 9)

So the common factor that can be cancelled is x−3.

lim_{x \to 3} \frac{x^{3} - 27}{x^{2} - 9} = lim_{x \to 3} \frac{x^{2} + 3 x + 9}{x + 3}

And now at this point, we can use the fact that this function is continuous at the point x=3 to substitute in 3.

lim_{x \to 3} \frac{x^{2} + 3 x + 9}{x + 3} = \frac{3^{2} + 9 + 9}{3 + 3} = \frac{9}{2}

Jeffrey Jordon

Expert2021-10-12Added 2605 answers

Answer is given below (on video)

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