Question

find all vertical asymptotes of the following function.

f(x)=\frac{2 x^{2}+7 x}{3 x^{2}+3 x - 90}

answer attempt 1 out of 2

no vertical asymptotes

Explanation:

Step1: Factor numerator and denominator

Numerator: \(2x^{2}+7x = x(2x + 7)\)
Denominator: \(3x^{2}+3x - 90=3(x^{2}+x - 30)=3(x + 6)(x - 5)\)
So \(f(x)=\frac{x(2x + 7)}{3(x + 6)(x - 5)}\)

Step2: Find values that make denominator zero

Set \(3(x + 6)(x - 5)=0\)
\(x+6 = 0\) gives \(x=-6\); \(x - 5=0\) gives \(x = 5\)

Step3: Check if numerator is zero at these values

When \(x=-6\), numerator \(=(-6)(2\times(-6)+7)=(-6)(-12 + 7)=(-6)\times(-5)=30
eq0\)
When \(x = 5\), numerator \(=5(2\times5+7)=5(10 + 7)=5\times17 = 85
eq0\)

Answer:

The vertical asymptotes are \(x=-6\) and \(x = 5\)