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To examine the horizontal asymptotes for the functions \( b) \ f(x) = 5 + x^2 e^x \) and \( c) \ f(x) = x + e^{-2x+1} \), let's proceed:

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### **(b) \( f(x) = 5 + x^2 e^x \):**

1. **As \( x \to \infty \):**
   - \( e^x \to \infty \) and \( x^2 \to \infty \), so \( x^2 e^x \to \infty \).
   - The function \( f(x) \to \infty \).
   - Hence, no horizontal asymptote exists for \( x \to \infty \).

2. **As \( x \to -\infty \):**
   - \( e^x \to 0^+ \) (since \( e^x \) approaches zero for \( x \to -\infty \)), and \( x^2 > 0 \), so \( x^2 e^x \to 0 \).
   - Therefore, \( f(x) \to 5 + 0 = 5 \).
   - **Horizontal asymptote: \( y = 5 \) as \( x \to -\infty \).**

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### **(c) \( f(x) = x + e^{-2x+1} \):**

1. **As \( x \to \infty \):**
   - In the term \( e^{-2x+1} \), the exponent \( -2x+1 \to -\infty \), so \( e^{-2x+1} \to 0^+ \).
   - The function \( f(x) \approx x + 0 = x \to \infty \).
   - Hence, no horizontal asymptote exists for \( x \to \infty \).

2. **As \( x \to -\infty \):**
   - The exponent \( -2x+1 \to \infty \), so \( e^{-2x+1} \to \infty \).
   - The term \( x \to -\infty \), but \( e^{-2x+1} \) dominates. Therefore, \( f(x) \to \infty \).
   - No horizontal asymptote exists for \( x \to -\infty \).

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### **Summary of Results:**
- For \( b) f(x) = 5 + x^2 e^x \), **horizontal asymptote: \( y = 5 \) as \( x \to -\infty \)**.
- For \( c) f(x) = x + e^{-2x+1} \), no horizontal asymptote exists.

Would you like further explanations or visualizations for these results? 

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### **Related Questions:**
1. What is the definition of a horizontal asymptote, and why does \( f(x) \) approach it as \( x \to \infty \) or \( x \to -\infty \)?
2. How can we determine horizontal asymptotes using limits algebraically?
3. Why does \( e^x \to \infty \) dominate polynomial growth such as \( x^2 \)?
4. Could \( c) f(x) = x + e^{-2x+1} \) have other asymptotes (e.g., oblique)?
5. How does \( e^{-2x+1} \) decay compared to other exponential terms like \( e^{-x} \)?

**Tip:** Always compute the limit of \( f(x) \) as \( x \to \pm\infty \) to identify horizontal asymptotes clearly.

Untersuchen Sie, ob der Graph von f für x → ∞ oder x → −∞ eine waagerechte Asymptote hat. Geben Sie in diesem Fall ihre Gleichung an. Analysieren Sie die Funktionen b) f(x) = 5 + x^2e^x und c) f(x) = x + e^(-2x+1).

Explore horizontal asymptotes for functions involving exponential growth and decay, specifically f(x) = 5 + x^2e^x and f(x) = x + e^(-2x+1). Learn how limits determine asymptotic behavior at infinity. Suitable for advanced high school students studying calculus and mathematical analysis.

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