How to find equations of vertical asymptotes?
Understand the Problem
The question is asking how to determine the equations of vertical asymptotes for a given function. Vertical asymptotes typically occur in rational functions where the denominator approaches zero while the numerator does not. We will find these asymptotes by setting the denominator equal to zero and solving for the variable.
Answer
Vertical asymptotes are found where $q(x) = 0$, excluding points where $p(x) = 0$.
Answer for screen readers
The vertical asymptotes occur at the values of $x$ that satisfy the equation $q(x) = 0$, provided the numerator does not equal zero at those values.
Steps to Solve
- Identify the Function
Start by identifying the rational function you are given. For example, if the function is $f(x) = \frac{p(x)}{q(x)}$, you need to focus on the denominator $q(x)$.
- Set the Denominator to Zero
To find the vertical asymptotes, set the denominator equal to zero:
$$ q(x) = 0 $$
- Solve for the Variable
Now, solve the equation you obtained in the previous step. This will give you the values of $x$ where the vertical asymptotes exist.
- Check the Numerator
To confirm that these values lead to vertical asymptotes, ensure that the numerator does not also equal zero at these points. If the numerator $p(x)$ is also zero at a certain value, then you may have a hole instead of a vertical asymptote.
The vertical asymptotes occur at the values of $x$ that satisfy the equation $q(x) = 0$, provided the numerator does not equal zero at those values.
More Information
Vertical asymptotes indicate the behavior of a function as it approaches certain values of $x$. They can lead to values that go to positive or negative infinity, defining the limits of the function's range.
Tips
- Failing to check if the numerator is also zero when factors equal zero in the denominator.
- Ignoring the fact that vertical asymptotes only occur where the denominator cancels out and the numerator does not equal zero.
