Graph all vertical and horizontal asymptotes of the rational function. $$ f(x)=\frac{4 x-13}{-2 x+9} $$

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Function by following steps:
- step0: Find the horizontal asymptotes:
  $$f\left(x\right)=\frac{4x-13}{-2x+9}$$
- step1: Evaluate the limits $$\lim _{x\rightarrow +\infty}\left(f\left(x\right)\right)$$ and $$\lim _{x\rightarrow -\infty}\left(f\left(x\right)\right):$$
  $$\begin{align}&\lim _{x\rightarrow +\infty}\left(\frac{4x-13}{-2x+9}\right)\&\lim _{x\rightarrow -\infty}\left(\frac{4x-13}{-2x+9}\right)\end{align}$$
- step2: Calculate:
  $$\begin{align}&-2\&-2\end{align}$$
- step3: The finite values are horizontal asymptotes:
  $$\begin{align}&f\left(x\right)=-2\end{align}$$
Find the vertical asymptotes of $$ f(x)=\frac{4 x-13}{-2 x+9} $$.
Function by following steps:
- step0: Find the vertical asymptotes:
  $$f\left(x\right)=\frac{4x-13}{-2x+9}$$
- step1: Evaluate the limit:
  $$\lim _{x\rightarrow \frac{9}{2}}\left(\frac{4x-13}{-2x+9}\right)$$
- step2: Evaluate the left-hand and the right-hand limits:
  $$\begin{align}&\lim _{x\rightarrow \frac{9}{2}^{-}}\left(\frac{4x-13}{-2x+9}\right)\&\lim _{x\rightarrow \frac{9}{2}^{+}}\left(\frac{4x-13}{-2x+9}\right)\end{align}$$
- step3: Evaluate the left-hand limit:
  $$\begin{align}&+\infty\&-\infty\end{align}$$
- step4: Limit does not exist:
  $$\textrm{The limit does not exist}$$
- step5: $$x=\frac{9}{2}$$ is a vertical asymptote$$:$$
  $$\begin{align}&x=\frac{9}{2}\textrm{ }\textrm{is a vertical asymptote}\end{align}$$
- step6: List all vertical asymptotes of the function:
  $$\begin{align}&x=\frac{9}{2}\end{align}$$
To find the asymptotes of the rational function

$$
f(x) = \frac{4x - 13}{-2x + 9},
$$

we have determined the following:

1. **Horizontal Asymptote**: The horizontal asymptote is given by the equation $$ f(x) = -2 $$.

2. **Vertical Asymptote**: The vertical asymptote occurs at $$ x = \frac{9}{2} $$.

### Summary of Asymptotes:
- **Horizontal Asymptote**: $$ y = -2 $$
- **Vertical Asymptote**: $$ x = \frac{9}{2} $$

These asymptotes can be graphed on the coordinate plane to visualize the behavior of the function as $$ x $$ approaches the vertical asymptote and as $$ y $$ approaches the horizontal asymptote.
The function has a horizontal asymptote at $$ y = -2 $$ and a vertical asymptote at $$ x = \frac{9}{2} $$.

Graph all vertical and horizontal asymptotes of the rational function. \[ f(x)=\frac{4 x-13}{-2 x+9} \]

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