Graph y=(3x^2-4x+1)/(2x^2+5x-3)

Find where the expression ${\displaystyle \frac{(3x-1)(x-1)}{(2x-1)(x+3)}}$ is undefined.

Since ${\displaystyle \frac{(3x-1)(x-1)}{(2x-1)(x+3)}}$${\displaystyle \to }$${\displaystyle \infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle -3}$ from the left and ${\displaystyle \frac{(3x-1)(x-1)}{(2x-1)(x+3)}}$${\displaystyle \to }$${\displaystyle -\infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle -3}$ from the right, then ${\displaystyle x=-3}$ is a vertical asymptote.

Since ${\displaystyle \frac{(3x-1)(x-1)}{(2x-1)(x+3)}}$${\displaystyle \to }$${\displaystyle \infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle \frac{1}{2}}$ from the left and ${\displaystyle \frac{(3x-1)(x-1)}{(2x-1)(x+3)}}$${\displaystyle \to }$${\displaystyle -\infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle \frac{1}{2}}$ from the right, then ${\displaystyle x=\frac{1}{2}}$ is a vertical asymptote.

List all of the vertical asymptotes:

Consider the rational function ${\displaystyle R\left(x\right)=\frac{a{x}^n}{b{x}^m}}$ where ${\displaystyle n}$ is the degree of the numerator and ${\displaystyle m}$ is the degree of the denominator.

Find ${\displaystyle n}$ and ${\displaystyle m}$.

Since ${\displaystyle n=m}$, the horizontal asymptote is the line ${\displaystyle y=\frac{a}{b}}$ where ${\displaystyle a=3}$ and ${\displaystyle b=2}$.

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

This is the set of all asymptotes.