Answer :
Absolutely! Let's thoroughly go through the process of identifying the vertical and horizontal asymptotes, as well as the end behavior, of the function [tex]\( f(x) = \frac{6x}{x - 36} \)[/tex].
### 1. Identifying Vertical Asymptotes:
Vertical asymptotes occur where the function approaches infinity or negative infinity as [tex]\( x \)[/tex] approaches a particular value, where the denominator equals zero.
For the function [tex]\( f(x) = \frac{6x}{x - 36} \)[/tex], we need to identify where the denominator is zero:
[tex]\[ x - 36 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 36 \][/tex]
So, the vertical asymptote is at [tex]\( x = 36 \)[/tex].
### 2. Identifying Horizontal Asymptotes:
Horizontal asymptotes are determined by analyzing the behavior of the function as [tex]\( x \)[/tex] approaches infinity or negative infinity.
For [tex]\( f(x) = \frac{6x}{x - 36} \)[/tex], let us analyze the limit as [tex]\( x \)[/tex] approaches infinity:
[tex]\[ \lim_{{x \to \infty}} \frac{6x}{x - 36} \][/tex]
Dividing the numerator and denominator by [tex]\( x \)[/tex]:
[tex]\[ \lim_{{x \to \infty}} \frac{6}{1 - \frac{36}{x}} \][/tex]
As [tex]\( x \to \infty \)[/tex], [tex]\(\frac{36}{x} \to 0\)[/tex]:
[tex]\[ \lim_{{x \to \infty}} \frac{6}{1 - 0} = 6 \][/tex]
So, the horizontal asymptote is [tex]\( y = 6 \)[/tex].
### 3. End Behavior:
The end behavior of the function is described by its behavior as [tex]\( x \)[/tex] approaches positive infinity and negative infinity.
#### As [tex]\( x \to \infty \)[/tex]:
Using the resolved horizontal asymptote:
[tex]\[ \lim_{{x \to \infty}} \frac{6x}{x - 36} = 6 \][/tex]
#### As [tex]\( x \to -\infty \)[/tex]:
We analyze the limit as [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ \lim_{{x \to -\infty}} \frac{6x}{x - 36} \][/tex]
Again, by dividing the numerator and denominator by [tex]\( x \)[/tex]:
[tex]\[ \lim_{{x \to -\infty}} \frac{6}{1 - \frac{36}{x}} \][/tex]
As [tex]\( x \to -\infty \)[/tex], [tex]\(\frac{36}{x} \to 0\)[/tex]:
[tex]\[ \lim_{{x \to -\infty}} \frac{6}{1 - 0} = 6 \][/tex]
So, as [tex]\( x \to -\infty \)[/tex], the function also approaches 6.
### Summary:
- Vertical Asymptote: [tex]\( x = 36 \)[/tex]
- Horizontal Asymptote: [tex]\( y = 6 \)[/tex]
- End Behavior: As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to 6 \)[/tex]. As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to 6 \)[/tex].
Hence, the vertical asymptote is at [tex]\( x = 36 \)[/tex], the horizontal asymptote is at [tex]\( y = 6 \)[/tex], and the end behavior indicates that the function approaches 6 as [tex]\( x \)[/tex] approaches either positive or negative infinity.
### 1. Identifying Vertical Asymptotes:
Vertical asymptotes occur where the function approaches infinity or negative infinity as [tex]\( x \)[/tex] approaches a particular value, where the denominator equals zero.
For the function [tex]\( f(x) = \frac{6x}{x - 36} \)[/tex], we need to identify where the denominator is zero:
[tex]\[ x - 36 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 36 \][/tex]
So, the vertical asymptote is at [tex]\( x = 36 \)[/tex].
### 2. Identifying Horizontal Asymptotes:
Horizontal asymptotes are determined by analyzing the behavior of the function as [tex]\( x \)[/tex] approaches infinity or negative infinity.
For [tex]\( f(x) = \frac{6x}{x - 36} \)[/tex], let us analyze the limit as [tex]\( x \)[/tex] approaches infinity:
[tex]\[ \lim_{{x \to \infty}} \frac{6x}{x - 36} \][/tex]
Dividing the numerator and denominator by [tex]\( x \)[/tex]:
[tex]\[ \lim_{{x \to \infty}} \frac{6}{1 - \frac{36}{x}} \][/tex]
As [tex]\( x \to \infty \)[/tex], [tex]\(\frac{36}{x} \to 0\)[/tex]:
[tex]\[ \lim_{{x \to \infty}} \frac{6}{1 - 0} = 6 \][/tex]
So, the horizontal asymptote is [tex]\( y = 6 \)[/tex].
### 3. End Behavior:
The end behavior of the function is described by its behavior as [tex]\( x \)[/tex] approaches positive infinity and negative infinity.
#### As [tex]\( x \to \infty \)[/tex]:
Using the resolved horizontal asymptote:
[tex]\[ \lim_{{x \to \infty}} \frac{6x}{x - 36} = 6 \][/tex]
#### As [tex]\( x \to -\infty \)[/tex]:
We analyze the limit as [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ \lim_{{x \to -\infty}} \frac{6x}{x - 36} \][/tex]
Again, by dividing the numerator and denominator by [tex]\( x \)[/tex]:
[tex]\[ \lim_{{x \to -\infty}} \frac{6}{1 - \frac{36}{x}} \][/tex]
As [tex]\( x \to -\infty \)[/tex], [tex]\(\frac{36}{x} \to 0\)[/tex]:
[tex]\[ \lim_{{x \to -\infty}} \frac{6}{1 - 0} = 6 \][/tex]
So, as [tex]\( x \to -\infty \)[/tex], the function also approaches 6.
### Summary:
- Vertical Asymptote: [tex]\( x = 36 \)[/tex]
- Horizontal Asymptote: [tex]\( y = 6 \)[/tex]
- End Behavior: As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to 6 \)[/tex]. As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to 6 \)[/tex].
Hence, the vertical asymptote is at [tex]\( x = 36 \)[/tex], the horizontal asymptote is at [tex]\( y = 6 \)[/tex], and the end behavior indicates that the function approaches 6 as [tex]\( x \)[/tex] approaches either positive or negative infinity.
