Question

In Exercises \(7-20,\) find the vertical asymptotes (if any) of the function. $$ g(x)=\frac{\frac{5}{2} x^{3}-x^{2}-4 x}{3 x^{2}-6 x-24} $$

Short answer

The vertical asymptotes of the function are \(x = 4\) and \(x = -2\).

Step-by-step solution

Factorize the denominator

To find where the denominator is zero, It's necessary to factorize the denominator which is a quadratic expression. Using the factor theorem, we rewrite \(3x^{2}-6x-24\) as \(3(x^{2}-2x-8)\), which can be further factorized into \(3(x - 4)(x + 2)\).

Solve for x in the denominator

The denominator is zero where \(x - 4= 0\) or \(x + 2= 0\). Solving these equations gives \(x = 4\) or \(x = -2\).

Check if these x values are roots of the numerator

The identified x values are roots of the entire function, and thus not vertical asymptotes, if they are roots of the numerator. To check this, substitute the x-values (4,-2) into the numerator equation (\(\frac{5}{2}x^{3}-x^{2}-4x\)). Both substitutions yield non-zero results, meaning the identified x-values are not roots of the numerator and hence are the vertical asymptotes.

State the final answer

The vertical asymptotes of the function are the lines \(x = 4\) and \(x = -2\).