Find the equations of any vertical asymptotes for the function below.

f left parenthesis x right parenthesis equals StartFraction x squared plus 6 x Over x cubed minus 49 x EndFraction

To find the equations of any vertical asymptotes for the function 

\[
f(x) = \frac{x^2 + 6x}{x^3 - 49x},
\]

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

### Step 1: Factor the Denominator
Let's start by factoring the denominator:

\[
x^3 - 49x = x(x^2 - 49)
\]

Notice that \(x^2 - 49\) is a difference of squares, so we can factor it further:

\[
x^2 - 49 = (x - 7)(x + 7)
\]

So, the denominator can be written as:

\[
x^3 - 49x = x(x - 7)(x + 7)
\]

### 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:

\[
x(x - 7)(x + 7) = 0
\]

This equation is zero when:

\[
x = 0
\]
\[
x - 7 = 0 \quad \Rightarrow \quad x = 7
\]
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\]

### Step 3: Check the Numerator
Now, let's check the numerator, \(x^2 + 6x\):

\[
x^2 + 6x = x(x + 6)
\]

The numerator is zero when:

\[
x = 0 \quad \text{or} \quad x = -6
\]

### Step 4: Identify Vertical Asymptotes
For vertical asymptotes, we need the denominator to be zero and the numerator to be non-zero at those points. 

- At \(x = 0\), both the numerator and denominator are zero, so \(x = 0\) does **not** produce a vertical asymptote (it might be a hole in the graph instead).
- At \(x = 7\), the numerator \(x(x + 6)\) is non-zero, so \(x = 7\) is a vertical asymptote.
- At \(x = -7\), the numerator \(x(x + 6)\) is also non-zero, so \(x = -7\) is a vertical asymptote.

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

\[
x = 7, \quad x = -7
\]

Would you like further 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. What happens to the function \(f(x)\) as \(x\) approaches each of these vertical asymptotes?
2. How can you determine if there is a hole at \(x = 0\) instead of an asymptote?
3. Can you find the horizontal asymptotes for this function, if any exist?
4. What is the significance of finding common factors in the numerator and denominator?
5. How do you graph functions with vertical asymptotes and holes?

**Tip:** Always check both the numerator and denominator when finding vertical asymptotes to rule out points that may actually be removable discontinuities (holes).

Learn how to find the equations of vertical asymptotes for the function f(x) = (x^2 + 6x) / (x^3 - 49x) using factorization and understanding polynomial functions. Detailed step-by-step solution provided.

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