Sure! Let's analyze the function [tex]\( f(x) = \frac{4x}{x-16} \)[/tex] to determine its asymptotes and end behavior.
### Vertical Asymptote
To find the vertical asymptote, we need to determine where the denominator of the function is equal to zero. Vertical asymptotes occur where the function is undefined due to division by zero.
[tex]\[ x - 16 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 16 \][/tex]
Thus, there is a vertical asymptote at [tex]\( x = 16 \)[/tex].
### Horizontal Asymptote
Next, let's analyze the horizontal asymptote. Horizontal asymptotes describe the behavior of a function as [tex]\( x \)[/tex] approaches positive or negative infinity.
To determine the horizontal asymptote for a rational function, we compare the degrees of the numerator and the denominator:
- If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
The numerator [tex]\( 4x \)[/tex] has a degree of 1, and the denominator [tex]\( x-16 \)[/tex] also has a degree of 1. The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 1.
The horizontal asymptote is given by the ratio of these leading coefficients:
[tex]\[ y = \frac{4}{1} = 4 \][/tex]
So, there is a horizontal asymptote at [tex]\( y = 4 \)[/tex].
### End Behavior
End behavior analysis involves understanding the behavior of the function as [tex]\( x \)[/tex] approaches positive or negative infinity.
To find the end behavior of [tex]\( f(x) \)[/tex], we will look at the limit of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex]:
[tex]\[ \lim_{{x \to \infty}} f(x) = \lim_{{x \to \infty}} \frac{4x}{x-16} \][/tex]
[tex]\[ = \frac{4(\infty)}{(\infty) - 16} \approx \frac{4\infty}{\infty} = 4 \][/tex]
Similarly,
[tex]\[ \lim_{{x \to -\infty}} f(x) = \lim_{{x \to -\infty}} \frac{4x}{x-16} \][/tex]
[tex]\[ = \frac{4(-\infty)}{(-\infty) - 16} \approx \frac{4(-\infty)}{-\infty} = 4 \][/tex]
As [tex]\( x \)[/tex] approaches both positive and negative infinity, [tex]\( f(x) \)[/tex] approaches 4. This confirms the horizontal asymptote and describes the end behavior.
### Summary
- Vertical asymptote: [tex]\( x = 16 \)[/tex]
- Horizontal asymptote: [tex]\( y = 4 \)[/tex]
- End behavior: As [tex]\( x \to \infty \)[/tex] or [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to 4 \)[/tex]
This is a detailed analysis of the asymptotes and end behavior of the function [tex]\( f(x) = \frac{4x}{x-16} \)[/tex].
