Evaluating Limits at Infinity of Exponential Functions: Calculus Tutorial

Analyzing Exponential Function Limits

Statement 1: Limit of 8 - e^(4x)

• 💡 The limit as x approaches infinity of 8 - e^(4x) is evaluated by considering the behavior of e^(4x).
• 📈 As x approaches infinity, e^(4x) approaches infinity, resulting in 8 minus infinity.
• ⚠️ Therefore, the limit of 8 - e^(4x) as x approaches infinity is negative infinity, making statement one true.

Statement 2: Limit of 1 / 3^x

• 🎯 The expression 1 / 3^x can be rewritten as 1 raised to the x over 3 raised to the x.
• 📉 As x approaches infinity, 3^x approaches infinity, and 1 over infinity approaches zero.
• ❌ Thus, statement two, which claims the limit is infinity, is false.

Statement 3: Limit of e^x / e^(2x)

• 🚀 Using exponent properties, e^x / e^(2x) simplifies to e^(x - 2x), which is e^(-2x).
• ➡️ The limit as x approaches infinity of e^(-2x) is equivalent to the limit of 1 / e^(2x).
• 🌟 As x approaches infinity, e^(2x) approaches infinity, and 1 over infinity is zero, making statement three true.

Conclusion

• ✅ Based on the analysis of each statement, statements one and three are true, while statement two is false.
• 📊 The final answer indicates that one and three are true.