Problem 4 - Calculus Early Transcendentals

Question

Where are the vertical asymptotes of a rational function located?

Step-by-Step Solution

Answer

Answer: 1. Define a rational function. 2. Identify the vertical asymptotes. 3. Set the denominator to 0. 4. Check for cancellation. 5. Finalize the vertical asymptotes.

Step 1: Define a rational function

A rational function is a function of the form R(x) = P(x)/Q(x) where P(x) and Q(x) are both polynomial functions. Vertical asymptotes occur at values of x that make the denominator, Q(x), equal to zero.

Step 2: Identify the vertical asymptotes

To determine where the vertical asymptotes are located, we need to find the values of x that make the denominator Q(x) equal to 0. These values of x are the vertical asymptotes because the function will become undefined (divide by zero) at these values.

Step 3: Setting the denominator to 0

Solve the equation Q(x) = 0 to find the x-values where the vertical asymptotes occur. Keep in mind that in some cases the function may not have any vertical asymptotes, depending upon the polynomials P(x) and Q(x).

Step 4: Check for cancellation

Before confirming the vertical asymptotes, make sure that none of the identified factors in Q(x) are also factors of P(x), as cancellation may remove the vertical asymptotes. If this is the case, then there will be no vertical asymptotes at that particular x-value.

Step 5: Finalize the vertical asymptotes

The x-values obtained from solving Q(x) = 0 (without any cancellation) represent the vertical asymptotes of the rational function R(x) = P(x)/Q(x).