To find the horizontal and vertical asymptotes of the given function:

$f(x) = 6 + \frac{7}{x-7}$

1. Horizontal Asymptote

• The horizontal asymptote describes the behavior of $f(x)$ as $x \to \infty$ or $x \to -\infty$.

• For the given function: $f(x) = 6 + \frac{7}{x-7}$ As $x \to \infty$ or $x \to -\infty$, the fraction $\frac{7}{x-7} \to 0$ because the denominator grows unbounded.

  Thus: $y = 6$ is the horizontal asymptote.

2. Vertical Asymptote

• The vertical asymptote occurs where the denominator of the fraction equals zero, as the function is undefined there.

  From the term $\frac{7}{x-7}$, set the denominator $x-7 = 0$: $x = 7$

  This is the vertical asymptote.

Final Answers:

• Horizontal Asymptote: $y = 6$
• Vertical Asymptote: $x = 7$

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Related Questions:

1. What are the asymptotes for the function $f(x) = \frac{x+2}{x-3}$?
2. How do you find horizontal asymptotes for rational functions in general?
3. What happens when there are no horizontal asymptotes?
4. Can a function have more than one vertical asymptote?
5. How can you graph a function with both vertical and horizontal asymptotes?

Tip:

To determine horizontal asymptotes, analyze the behavior of $f(x)$ as $x \to \infty$ or $x \to -\infty$.