Question

question
find all vertical asymptotes of the following function.
f(x)=\frac{2 x^{2}-20 x}{3 x-30}
answer attempt 1 out of 2

Explanation:

Step1: Simplify the function

Factor numerator and denominator:
Numerator: \(2x^{2}-20x = 2x(x - 10)\)
Denominator: \(3x-30=3(x - 10)\)
So \(f(x)=\frac{2x(x - 10)}{3(x - 10)}=\frac{2x}{3},x
eq10\)

Step2: Analyze vertical asymptotes

A vertical asymptote occurs at \(x = a\) if \(\lim_{x
ightarrow a^{-}}f(x)=\pm\infty\) or \(\lim_{x
ightarrow a^{+}}f(x)=\pm\infty\).
Since \(f(x)=\frac{2x}{3}\) (after canceling non - zero factors for \(x
eq10\)), and the function \(y = \frac{2x}{3}\) is a linear function (a polynomial of degree 1). The only point of discontinuity is \(x = 10\), but \(\lim_{x
ightarrow10}f(x)=\frac{2\times10}{3}=\frac{20}{3}\) (a finite value).

Answer:

No Vertical Asymptotes