For the function $$ F(x)=\frac{x^{2}-3}{x^{2}} $$, (a) graph the rational function: asymptotes.

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Function by following steps:
- step0: Find the horizontal asymptotes:
  $$F\left(x\right)=\frac{x^{2}-3}{x^{2}}$$
- step1: Evaluate the limits $$\lim _{x\rightarrow +\infty}\left(F\left(x\right)\right)$$ and $$\lim _{x\rightarrow -\infty}\left(F\left(x\right)\right):$$
  $$\begin{align}&\lim _{x\rightarrow +\infty}\left(\frac{x^{2}-3}{x^{2}}\right)\&\lim _{x\rightarrow -\infty}\left(\frac{x^{2}-3}{x^{2}}\right)\end{align}$$
- step2: Calculate:
  $$\begin{align}&1\&1\end{align}$$
- step3: The finite values are horizontal asymptotes:
  $$\begin{align}&F\left(x\right)=1\end{align}$$
Find the oblique asymptotes of $$ F(x)=(x^2-3)/(x^2) $$.
Function by following steps:
- step0: Find the oblique asymptotes:
  $$F\left(x\right)=\frac{x^{2}-3}{x^{2}}$$
- step1: Evaluate the limits $$\lim _{x\rightarrow +\infty}\left(\frac{F\left(x\right)}{x}\right)$$ and $$\lim _{x\rightarrow -\infty}\left(\frac{F\left(x\right)}{x}\right):$$
  $$\begin{align}&\lim _{x\rightarrow +\infty}\left(\frac{\frac{x^{2}-3}{x^{2}}}{x}\right)\&\lim _{x\rightarrow -\infty}\left(\frac{\frac{x^{2}-3}{x^{2}}}{x}\right)\end{align}$$
- step2: Calculate:
  $$\begin{align}&0\&0\end{align}$$
- 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:
  $$\textrm{No oblique asymptotes}$$
Find the vertical asymptotes of $$ F(x)=(x^2-3)/(x^2) $$.
Function by following steps:
- step0: Find the vertical asymptotes:
  $$F\left(x\right)=\frac{x^{2}-3}{x^{2}}$$
- step1: Evaluate the limit:
  $$\lim _{x\rightarrow 0}\left(\frac{x^{2}-3}{x^{2}}\right)$$
- step2: Evaluate the left-hand and the right-hand limits:
  $$\begin{align}&\lim _{x\rightarrow 0^{-}}\left(\frac{x^{2}-3}{x^{2}}\right)\&\lim _{x\rightarrow 0^{+}}\left(\frac{x^{2}-3}{x^{2}}\right)\end{align}$$
- step3: Evaluate the left-hand limit:
  $$\begin{align}&-\infty\&-\infty\end{align}$$
- step4: Calculate:
  $$-\infty$$
- step5: $$x=0$$ is a vertical asymptote$$:$$
  $$\begin{align}&x=0\textrm{ }\textrm{is a vertical asymptote}\end{align}$$
- step6: List all vertical asymptotes of the function:
  $$\begin{align}&x=0\end{align}$$
For the function $$ F(x) = \frac{x^{2}-3}{x^{2}} $$, we can determine the asymptotes as follows:

### Horizontal Asymptote
- The horizontal asymptote is found by evaluating the limit of $$ F(x) $$ as $$ x $$ approaches infinity. In this case, as $$ x \to \infty $$ or $$ x \to -\infty $$, the function approaches:
  $$
  F(x) \to 1
  $$
  Therefore, the horizontal asymptote is:
  $$
  y = 1
  $$

### Vertical Asymptote
- The vertical asymptote occurs where the denominator is zero (and the numerator is not zero). Here, the denominator $$ x^{2} $$ is zero when:
  $$
  x = 0
  $$
  Thus, there is a vertical asymptote at:
  $$
  x = 0
  $$

### Oblique Asymptote
- There are no oblique asymptotes for this function since the degree of the numerator is not greater than the degree of the denominator.

### Summary of Asymptotes
- **Horizontal Asymptote:** $$ y = 1 $$
- **Vertical Asymptote:** $$ x = 0 $$
- **Oblique Asymptote:** None

These asymptotes will help in graphing the rational function $$ F(x) $$.
The function $$ F(x) = \frac{x^{2}-3}{x^{2}} $$ has a horizontal asymptote at $$ y = 1 $$ and a vertical asymptote at $$ x = 0 $$. There are no oblique asymptotes.

For the function \( F(x)=\frac{x^{2}-3}{x^{2}} \), (a) graph the rational function: asymptotes.

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