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📝 Limits at Infinity Rigorous

📖 From Calculus • 2. Limits and Continuity an Introduction • 34 questions available

Practice MCQs for Limits at Infinity Rigorous. Test your knowledge with carefully crafted questions across easy, medium, and hard difficulty levels.

🔄 Last updated: 2026-07-12

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Q1. What is the rigorous definition of \(\lim_{x\to +\infty} f(x) = L\)?

🔹 A. For all \(\epsilon>0\), there exists \(N>0\) such that \(|f(x)-L|<\epsilon\) if \(x>N\)

🔹 B. For all \(\epsilon>0\), there exists \(N>0\) such that \(|f(x)-L|<\epsilon\) if \(x<N\)

🔹 C. For all \(N>0\), there exists \(\epsilon>0\) such that \(|f(x)-L|<\epsilon\) if \(x>N\)

🔹 D. For all \(\epsilon>0\), there exists \(N>0\) such that \(|f(x)-L|>\epsilon\) if \(x>N\)

💡 Difficulty: easy | ✅ Correct: A

Q2. In Example 4, proving \(\lim_{x\to +\infty} \frac{1}{x} = 0\), what value of \(N\) is chosen?

🔹 A. \(N = \epsilon\)

🔹 B. \(N = \frac{1}{\epsilon}\)

🔹 C. \(N = \epsilon^2\)

🔹 D. \(N = \sqrt{\epsilon}\)

💡 Difficulty: medium | ✅ Correct: B

Q3. What is the rigorous definition of \(\lim_{x\to -\infty} f(x) = L\)?

🔹 A. For all \(\epsilon>0\), there exists \(N>0\) such that \(|f(x)-L|<\epsilon\) if \(x>N\)

🔹 B. For all \(\epsilon>0\), there exists \(N<0\) such that \(|f(x)-L|<\epsilon\) if \(x<N\)

🔹 C. For all \(N>0\), there exists \(\epsilon>0\) such that \(|f(x)-L|<\epsilon\) if \(x<N\)

🔹 D. For all \(\epsilon>0\), there exists \(N<0\) such that \(|f(x)-L|>\epsilon\) if \(x<N\)

💡 Difficulty: easy | ✅ Correct: B

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🔗 Related Topics

📝 Approximating Roots 📝 Areas and Limits 📝 Continuity in Applications 📝 Continuity of Compositions 📝 Continuity of Inverse Functions

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