Question

Find the vertical and horizontal asymptotes. Write the asymptotes as equations of lines. \(f(x)=\frac{x^{2}+1}{x^{3}-8}\)

Short answer

The vertical asymptote of the function is \(x = 2\) and the horizontal asymptote is \(y = 0\).

Step-by-step solution

Identify Vertical Asymptotes

Vertical asymptotes occur when the denominator equals 0. Solve the equation \(x^3 - 8 = 0\).

Solve for Vertical Asymptotes

The equation in step 1 can be rewritten as \(x^3 = 8\), and solving this gives \(x = 2\). Thus, there is one vertical asymptote, and it is \(x = 2\).

Identify Horizontal Asymptotes

Identify the horizontal asymptote. Looking at the degrees of the numerator and denominator, we see that the denominator's degree (3) is larger. Thus, the horizontal asymptote is \(y = 0\).

Key concepts

Vertical Asymptotes

Vertical asymptotes are the values for which a function approaches infinity or negative infinity as the input approaches a specific value. More explicitly, they can be found where the denominator of a fraction becomes zero, because a non-zero number divided by zero tends to infinity.
In our exercise, we had the function
\(f(x)=\frac{x^{2}+1}{x^{3}-8}\).
To find vertical asymptotes, we set the denominator equal to zero and solve for x. Here,
\(x^{3}-8=0\) simplifies to
\(x^3 = 8\), and the value we find is
\(x = 2\).
This gives us the vertical asymptote as an equation of a line:
\(x=2\). Any graph of this function will show that as x approaches 2, f(x) shoots off to infinity. It's important to visualize this as a line that the graph of the function will approach but never touch.

Horizontal Asymptotes

In contrast to vertical asymptotes, horizontal asymptotes describe the behavior of a function as the input grows without bound. They are associated with the end behavior of the function. To find a horizontal asymptote, examine the degrees of the numerator and denominator polynomials.
If the degree of the numerator is lower than the degree of the denominator, as in our function \(f(x)=\frac{x^{2}+1}{x^{3}-8}\), the horizontal asymptote is the x-axis, which is the line
\(y=0\). If the numerator's degree is higher or equal, different rules apply, but no matter what, the horizontal asymptote represents the value that y approaches as x becomes very large or very small. It provides a 'horizon' for the graph.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example is
\(x^2 + 2x + 1\).
The important thing to remember about polynomials is that they are continuous and smooth functions - there are no sharp turns or holes in their graphs. In the exercise, our function's denominator
\(x^3 - 8\) is a polynomial that's being used to determine the vertical asymptote. Polynomial functions usually have well-defined behaviors, and this predictability helps when identifying features like asymptotes.

Limits of Functions

Limits are fundamental to calculus and mathematical analysis and pertain to the behavior of a function as it approaches a particular point. The concept of a limit helps us understand the value that a function approaches as the input gets closer to a specific number.
When finding asymptotes, we are essentially looking for limits. For a vertical asymptote at
\(x=2\), we would examine the limit of \(f(x)\) as x approaches 2. In the case of horizontal asymptotes, we seek the limit of \(f(x)\) as x approaches infinity or negative infinity. With limits, we can rigorously define those points at which functions don't behave 'nicely', such as where they might approach infinity, or where there might not be an actual value for the function.