To determine the asymptotes of the function [tex]\( f(x) = \frac{5}{x+7} - 8 \)[/tex], we need to identify the vertical and horizontal asymptotes.
### Vertical Asymptote
1. Find when the denominator is zero:
For a vertical asymptote, set the denominator of the rational function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x + 7 = 0
\][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[
x = -7
\][/tex]
So, [tex]\( x = -7 \)[/tex] is a vertical asymptote of the function.
### Horizontal Asymptote
2. Examine the limit as [tex]\( x \)[/tex] approaches infinity:
For the horizontal asymptote, you need to consider the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches infinity or negative infinity. In this case, notice that the term [tex]\( \frac{5}{x+7} \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches ±∞. Therefore, the horizontal asymptote is determined by the constant term outside the fraction:
[tex]\[
y = -8
\][/tex]
So, [tex]\( y = -8 \)[/tex] is a horizontal asymptote of the function.
### Summary
- The vertical asymptote is given by [tex]\( x = -7 \)[/tex].
- The horizontal asymptote is given by [tex]\( y = -8 \)[/tex].
Thus, the correct answers are:
- [tex]\( y = -8 \)[/tex]
- [tex]\( x = -7 \)[/tex]
