To find the asymptotes of the function [tex]\( f(x) = \frac{3}{x-7} + 2 \)[/tex], we need to analyze the behavior of the function as [tex]\( x \)[/tex] approaches certain critical values. There are two types of asymptotes to consider: vertical asymptotes and horizontal asymptotes.
### Vertical Asymptote:
1. Identify the vertical asymptote by examining the denominator:
The vertical asymptote occurs where the denominator of the rational expression is zero because at this point, the function tends towards infinity or negative infinity.
For the expression [tex]\(\frac{3}{x-7}\)[/tex], the denominator is zero when [tex]\( x-7 = 0 \)[/tex].
Solving for [tex]\( x \)[/tex]:
[tex]\[
x - 7 = 0
\][/tex]
[tex]\[
x = 7
\][/tex]
Hence, the vertical asymptote is at:
[tex]\[
x = 7
\][/tex]
### Horizontal Asymptote:
2. Determine the horizontal asymptote by evaluating the behavior of the function as [tex]\( x \)[/tex] approaches infinity or negative infinity:
The horizontal asymptote is found by analyzing the limits of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] tends towards infinity.
As [tex]\( x \to \infty \)[/tex] or [tex]\( x \to -\infty \)[/tex], the term [tex]\(\frac{3}{x-7}\)[/tex] approaches 0 because the numerator is a constant and the denominator grows without bound. Thus, the fraction [tex]\(\frac{3}{x-7}\)[/tex] becomes insignificant and the function approaches the constant term remaining:
[tex]\[
\lim_{{x \to \infty}} \left( \frac{3}{x-7} + 2 \right) = 2
\][/tex]
[tex]\[
\lim_{{x \to -\infty}} \left( \frac{3}{x-7} + 2 \right) = 2
\][/tex]
Therefore, the horizontal asymptote is:
[tex]\[
y = 2
\][/tex]
### Conclusion:
- Vertical Asymptote: [tex]\( x = 7 \)[/tex]
- Horizontal Asymptote: [tex]\( y = 2 \)[/tex]
