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the equations of the vertical asymptotes (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)


The graph you provided shows a function \( f(x) \) with several vertical asymptotes. Vertical asymptotes occur where the function approaches infinity, typically when the denominator of a rational function equals zero and the numerator is non-zero.

From the graph, vertical asymptotes can be identified where the function sharply increases or decreases without bound (where it shoots off to \( +\infty \) or \( -\infty \)). These points often indicate the x-values where the function is undefined.

Looking at the graph, there is a clear vertical asymptote at \( x = 0 \), where the function sharply increases and decreases around that point.

Thus, the equation for the vertical asymptote is:
\[
x = 0
\]

If you notice any other vertical asymptotes, feel free to let me know! Would you like more details on identifying these features or further explanation?

Here are five related questions to expand on this topic:
1. What are the horizontal asymptotes of this function, if any?
2. How can you determine asymptotes from a rational function algebraically?
3. Are there any points of discontinuity on this graph?
4. What is the significance of asymptotes in real-world applications?
5. Can a function have both vertical and horizontal asymptotes?

**Tip:** To find vertical asymptotes algebraically, solve for the points where the denominator of the function is zero while ensuring the numerator isn't zero at those points.

What are the equations of the vertical asymptotes?

Learn how to find the vertical asymptotes from a graph. This problem involves analyzing the function graph and identifying key asymptotic behaviors. Suitable for students in Grades 10-12, covering concepts like limits and asymptotes in rational functions.

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