Question

Find the vertical and horizontal asymptotes. Write the asymptotes as equations of lines. \(f(x)=\frac{4}{(x-2)^{3}}\)

Short answer

The vertical asymptote is given by the line \(x = 2\) and the horizontal asymptote is \(y = 0\).

Step-by-step solution

Compute vertical asymptotes

Vertical asymptotes happen when the denominator of a rational function goes to zero. Therefore, set the denominator \(x-2\) to zero and solve for \(x\), which gives \(x = 2\). So, the vertical asymptote is \(x = 2\).

Compute horizontal asymptotes

For finding horizontal asymptotes, it's necessary to calculate the limit of the function as \(x\) approaches positive and negative infinity. Given function is \(f(x)=\frac{4}{(x-2)^{3}}\). As \(x\) goes to positive and negative infinity, the value of function \(f\) is going to zero since \(x\) is cubed in the denominator. Thus, the horizontal asymptote is \(y = 0\).

Key concepts

Vertical Asymptotes

When you encounter vertical asymptotes, you're dealing with the points where a rational function becomes undefined. In simpler terms, these are the values of \(x\) that cause the denominator of the function to equal zero.
This results in values at which the function shoots off to infinity, either positively or negatively.
In our specific case, the rational function is \(f(x) = \frac{4}{(x-2)^3}\).
Here is how you find the vertical asymptote:

• Look at the denominator: \((x-2)^3\), and set it equal to zero: \(x-2 = 0\).
• Solve for \(x\): \(x = 2\).
• Thus, the vertical asymptote is at the line \(x = 2\).

Remember, vertical asymptotes occur where the denominator can't compensate to make the function defined. Instead, it tends toward infinity or negative infinity as it approaches the asymptote line.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \(x\) approaches either positive or negative infinity. These asymptotes show us how a function behaves far out in either direction. They tell us if the function stabilizes or flattens out as it spreads across large values of \(x\).
For the given function \(f(x) = \frac{4}{(x-2)^3}\), to find the horizontal asymptote:

• Examine the degrees of the numerator and the denominator.
• The numerator here is a constant, which means it can be treated as a polynomial of degree 0.
• The degree of the denominator is 3, due to the \((x-2)^3\) term.

Since the degree of the denominator is greater than that of the numerator, as \(x\) approaches infinity or negative infinity, \(f(x)\) will approach zero.
Thus, the horizontal asymptote is the line \(y = 0\).
Keep in mind that horizontal asymptotes represent the end behavior of the function, not necessarily where the function can't exist.

Rational Functions

Rational functions are those which are the ratio of two polynomials.
They can be expressed in the form \(f(x) = \frac{p(x)}{q(x)}\), where both \(p(x)\) and \(q(x)\) are polynomials.
These functions can exhibit various features such as vertical and horizontal asymptotes, where their behavior is particularly noteworthy.
Let's look at some characteristics of rational functions:

• They are defined everywhere except where the denominator is zero.
• Vertical asymptotes occur where the function is undefined.
• Horizontal asymptotes describe the behavior of the function at the extremes, far to the left or right on the graph.

The example problem you worked on, \(f(x) = \frac{4}{(x-2)^3}\), is a classic example of a simple rational function, having a significant impact due to the powers at play in its denominator and the constant nature of its numerator.
Such examples highlight the ability of rational functions to model various phenomena with asymptotic behavior.