Find the equations of any vertical asymptotes.
​f(x)equalsStartFraction x squared plus 1 Over left parenthesis x squared minus 4 right parenthesis left parenthesis x squared minus 16 right parenthesis EndFraction

To find the equations of any vertical asymptotes of the function 

\[
f(x) = \frac{x^2 + 1}{(x^2 - 4)(x^2 - 16)},
\]

we need to determine the values of \(x\) that make the denominator equal to zero, as these points are where the function may have vertical asymptotes.

### Step 1: Factor the Denominator
First, let's factor the expressions in the denominator:

\[
x^2 - 4 = (x - 2)(x + 2)
\]
\[
x^2 - 16 = (x - 4)(x + 4)
\]

So the denominator can be written as:

\[
(x^2 - 4)(x^2 - 16) = (x - 2)(x + 2)(x - 4)(x + 4)
\]

### Step 2: Set the Denominator Equal to Zero
The vertical asymptotes occur at the values of \(x\) where the denominator is zero and the numerator is non-zero. Therefore, we need to solve the equation:

\[
(x - 2)(x + 2)(x - 4)(x + 4) = 0
\]

This equation is zero when:

\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]

### Step 3: Check the Numerator
The numerator \(x^2 + 1\) is never zero for real values of \(x\) because \(x^2 + 1 \geq 1\) for all real \(x\). Therefore, at the points \(x = 2\), \(x = -2\), \(x = 4\), and \(x = -4\), the numerator is non-zero.

### Conclusion:
The vertical asymptotes of the function are at:

\[
x = 2, \quad x = -2, \quad x = 4, \quad x = -4
\]

Would you like more details on how to analyze the behavior of the function near these asymptotes, or do you have any questions?

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Here are five related questions:
1. How do horizontal asymptotes differ from vertical asymptotes?
2. Can you find horizontal or oblique asymptotes for this function?
3. What happens to the function \(f(x)\) as \(x\) approaches infinity?
4. What is the significance of the numerator in determining vertical asymptotes?
5. How can we determine if a vertical asymptote is a simple or higher-order pole?

**Tip:** Always check if the numerator of a rational function is zero at the points where the denominator is zero to rule out any removable discontinuities.

Learn how to find the equations of vertical asymptotes for the rational function f(x) = (x^2 + 1) / ((x^2 - 4)(x^2 - 16)). Detailed step-by-step solution provided, including factoring the denominator and checking the numerator. Suitable for high school students studying rational functions.

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