Find theand vertical asymptotes of​ f(x).

​f(x)equals=StartFraction x Superscript 4 Over x squared plus 2 EndFraction

Given the function:

\[
f(x) = \frac{x^4}{x^2 + 2}
\]

To find the vertical asymptotes, we need to determine the values of \( x \) that make the denominator equal to zero. Vertical asymptotes occur where the denominator is zero and the numerator is non-zero.

### Step 1: Set the denominator equal to zero

\[
x^2 + 2 = 0
\]

### Step 2: Solve for \( x \)

\[
x^2 = -2
\]

\[
x = \sqrt{-2}
\]

Since the square root of a negative number is not a real number, there are no real values of \( x \) that make the denominator zero. Therefore, the function does not have any vertical asymptotes.

### Conclusion:
The correct answer is:
**B. There are no vertical asymptotes.**

Would you like further details, or do you have any questions?

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Here are 5 related questions for further exploration:
1. What are horizontal asymptotes and how do you find them?
2. How can you determine the behavior of \( f(x) \) near vertical asymptotes?
3. What is the significance of a hole in a function?
4. How do you find oblique asymptotes in a rational function?
5. What happens if both the numerator and denominator equal zero at the same point?

**Tip:** When dealing with rational functions, always check if the denominator can be zero to find vertical asymptotes.

Learn how to find vertical asymptotes of the function f(x) = x^4 / (x^2 + 2) with a step-by-step solution. Understand the concept of vertical asymptotes in rational functions and apply it to this specific example. Suitable for high school students studying advanced algebra.

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