Graph all asymptotes of the rational function. $$ f(x)=\frac{-x^{2}-2 x+3}{x-2} $$

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Function by following steps:
- step0: Find the horizontal asymptotes:
  $$f\left(x\right)=-x^{2}-2x+\frac{3}{x-2}$$
- step1: Evaluate the limits $$\lim _{x\rightarrow +\infty}\left(f\left(x\right)\right)$$ and $$\lim _{x\rightarrow -\infty}\left(f\left(x\right)\right):$$
  $$\begin{align}&\lim _{x\rightarrow +\infty}\left(-x^{2}-2x+\frac{3}{x-2}\right)\&\lim _{x\rightarrow -\infty}\left(-x^{2}-2x+\frac{3}{x-2}\right)\end{align}$$
- step2: Calculate:
  $$\begin{align}&-\infty\&-\infty\end{align}$$
- step3: The function has no horizontal asymptotes:
  $$\textrm{No horizontal asymptotes}$$
Find the vertical asymptotes of $$ f(x)=-x^{2}-2 x+3/(x-2) $$.
Function by following steps:
- step0: Find the vertical asymptotes:
  $$f\left(x\right)=-x^{2}-2x+\frac{3}{x-2}$$
- step1: Evaluate the limit:
  $$\lim _{x\rightarrow 2}\left(-x^{2}-2x+\frac{3}{x-2}\right)$$
- step2: Calculate:
  $$\textrm{The limit does not exist}$$
- step3: $$x=2$$ is a vertical asymptote$$:$$
  $$\begin{align}&x=2\textrm{ }\textrm{is a vertical asymptote}\end{align}$$
- step4: List all vertical asymptotes of the function:
  $$\begin{align}&x=2\end{align}$$
Find the oblique asymptotes of $$ f(x)=-x^{2}-2 x+3/(x-2) $$.
Function by following steps:
- step0: Find the oblique asymptotes:
  $$f\left(x\right)=-x^{2}-2x+\frac{3}{x-2}$$
- step1: Evaluate the limits $$\lim _{x\rightarrow +\infty}\left(\frac{f\left(x\right)}{x}\right)$$ and $$\lim _{x\rightarrow -\infty}\left(\frac{f\left(x\right)}{x}\right):$$
  $$\begin{align}&\lim _{x\rightarrow +\infty}\left(\frac{-x^{2}-2x+\frac{3}{x-2}}{x}\right)\&\lim _{x\rightarrow -\infty}\left(\frac{-x^{2}-2x+\frac{3}{x-2}}{x}\right)\end{align}$$
- step2: Calculate:
  $$\begin{align}&-\infty\&+\infty\end{align}$$
- step3: The function has no oblique asymptote:
  $$\begin{align}&\textrm{No oblique asymptotes}\&\textrm{No oblique asymptotes}\end{align}$$
- step4: The function has no oblique asymptote:
  $$\textrm{No oblique asymptotes}$$
To analyze the rational function

$$
f(x) = \frac{-x^{2}-2x+3}{x-2},
$$

we will identify the asymptotes step by step.

### 1. Horizontal Asymptotes
For rational functions, horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator:

- The degree of the numerator $$-x^2 - 2x + 3$$ is 2.
- The degree of the denominator $$x - 2$$ is 1.

Since the degree of the numerator is greater than the degree of the denominator, there are **no horizontal asymptotes**.

### 2. Vertical Asymptotes
Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points).

Setting the denominator equal to zero:

$$
x - 2 = 0 \implies x = 2.
$$

Thus, there is a vertical asymptote at

$$
x = 2.
$$

### 3. Oblique Asymptotes
Oblique (or slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. In this case, since the degree of the numerator (2) is greater than the degree of the denominator (1) by more than one, there are **no oblique asymptotes**.

### Summary of Asymptotes
- **Horizontal Asymptotes:** None
- **Vertical Asymptotes:** $$x = 2$$
- **Oblique Asymptotes:** None

These findings indicate that the function has a vertical asymptote at $$x = 2$$ and no horizontal or oblique asymptotes.
The function $$ f(x) = \frac{-x^{2}-2x+3}{x-2} $$ has a vertical asymptote at $$ x = 2 $$ and no horizontal or oblique asymptotes.

Graph all asymptotes of the rational function. \[ f(x)=\frac{-x^{2}-2 x+3}{x-2} \]

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