https://file.aiquickdraw.com/mathbot/file/173025702051424piostr.jpg

Here is the graph of the function \( f(x) = \frac{2x}{\sqrt{x^2 + 2}} \). The function shows a smooth curve that approaches \( f(x) = 2 \) as \( x \to +\infty \) and \( f(x) = -2 \) as \( x \to -\infty \). The function crosses the origin at \( x = 0 \), where \( f(0) = 0 \).

Would you like a further breakdown of the graph’s behavior, or do you have any questions?

Here are some related questions you might find useful:
1. What are the asymptotes of this function?
2. How does the derivative of \( f(x) \) behave, and what does it tell us about the slope of the curve?
3. How does the function behave as \( x \) approaches infinity and negative infinity?
4. Can we find the critical points of this function?
5. How would you find the domain and range of this function?

**Tip:** For functions with square roots in the denominator, always check for any potential undefined points and consider the behavior as \( x \) approaches infinity.

Explore the graph of the function f(x) = 2x / sqrt(x^2 + 2). This function demonstrates asymptotic behavior, approaching limits as x approaches positive and negative infinity, and crossing the origin at x = 0. Ideal for high school students studying calculus concepts like function graphing and limits.

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