1.7.1: Resources and Key Concepts

Resources

Videos

• Finite limits at infinity
  • Limits at Infinity - Evaluating Finite Limits at Infinity
• Proving infinite limits
  • Limits at Infinity - Proving Infinite Limits at Infinity
• Proving limits at infinity
  • Limits at Infinity - Proving Finite Limits at Infinity
• Calculus proofs for limits
  • Limits at Infinity - A Summary of Limit Proofs

Key Concepts

Definitions

• Infinite Limit at a Finite Number (Precise Definition): Let \(f(x)\) be defined for all \(x \neq a\) over an open interval containing \(a\).
  • We say \( \displaystyle \lim_{x \to a} f(x) = \infty\) if for every \(N \gg 0\), there exists a \(\delta > 0\), such that if \(0 < |x - a| < \delta\), then \(f(x) > N\). (Symbolically: \(\forall N \gg 0, \exists \delta > 0 \ni 0 < |x-a| < \delta \Rightarrow f(x) > N\))
  • We say \( \displaystyle \lim_{x \to a} f(x) = -\infty\) if for every \(N \ll 0\), there exists a \(\delta > 0\), such that if \(0 < |x - a| < \delta\), then \(f(x) < N\). (Symbolically: \(\forall N \ll 0, \exists \delta > 0 \ni 0 < |x-a| < \delta \Rightarrow f(x) < N\))
• Vertical Asymptote: The vertical line \(x=a\) is called a vertical asymptote of the function \(f(x)\) if at least one of the following is true: \( \displaystyle \lim_{x \to a} f(x) = \infty\), \( \displaystyle \lim_{x \to a} f(x) = -\infty\), \( \displaystyle \lim_{x \to a^+} f(x) = \infty\), \( \displaystyle \lim_{x \to a^+} f(x) = -\infty\), \( \displaystyle \lim_{x \to a^-} f(x) = \infty\), or \( \displaystyle \lim_{x \to a^-} f(x) = -\infty\).
• Finite Limit at Infinity (Precise Definition):
  • Let \(f(x)\) be defined for all \(x > a\). We say \( \displaystyle \lim_{x \to \infty} f(x) = L\) if for every \(\epsilon > 0\), there exists a number \(M > 0\), such that if \(x > M\), then \(|f(x) - L| < \epsilon\). (Symbolically: \(\forall \epsilon > 0, \exists M > 0 \ni x > M \Rightarrow |f(x)-L| < \epsilon\))
  • Let \(f(x)\) be defined for all \(x < a\). We say \( \displaystyle \lim_{x \to -\infty} f(x) = L\) if for every \(\epsilon > 0\), there exists a number \(M < 0\), such that if \(x < M\), then \(|f(x) - L| < \epsilon\). (Symbolically: \(\forall \epsilon > 0, \exists M < 0 \ni x < M \Rightarrow |f(x)-L| < \epsilon\))
• Horizontal Asymptote: The line \(y=L\) is called a horizontal asymptote of the function \(f(x)\) if at least one of the following is true: \( \displaystyle \lim_{x \to \infty} f(x) = L\) or \( \displaystyle \lim_{x \to -\infty} f(x) = L\).
• Infinite Limit at Infinity (Precise Definition):
  • As \(x \to \infty\): Let \(f(x)\) be defined for all \(x > a\).
    • We say \( \displaystyle \lim_{x \to \infty} f(x) = \infty\) if for every \(N \gg 0\), there exists a number \(M > 0\), such that if \(x > M\), then \(f(x) > N\).
    • We say \( \displaystyle \lim_{x \to \infty} f(x) = -\infty\) if for every \(N \ll 0\), there exists a number \(M > 0\), such that if \(x > M\), then \(f(x) < N\).
  • As \(x \to -\infty\): Let \(f(x)\) be defined for all \(x < a\).
    • We say \( \displaystyle \lim_{x \to -\infty} f(x) = \infty\) if for every \(N \gg 0\), there exists a number \(M < 0\), such that if \(x < M\), then \(f(x) > N\).
    • We say \( \displaystyle \lim_{x \to -\infty} f(x) = -\infty\) if for every \(N \ll 0\), there exists a number \(M < 0\), such that if \(x < M\), then \(f(x) < N\).

Theorems

• Theorem: \( \displaystyle \lim_{x \to \infty} e^{-x} = 0\).
• Corollary: If \(k > 0\), then \( \displaystyle \lim_{x \to \infty} e^{-kx} = 0\).
• Theorem: If \(p > 0\) is a rational number, then \( \displaystyle \lim_{x \to \infty} \frac{1}{x^p} = 0\). (Also stated: If \(p > 0\) is a rational number such that \(x^p\) is defined for all \(x\), then \( \displaystyle \lim_{x \to -\infty} \frac{1}{x^p} = 0\). The proof for \(x \to -\infty\) is left as a homework exercise.)

Common Mistakes

• Inequality Direction Switch: Forgetting that an inequality switches directions if you multiply or divide by a negative number.
• Even Powers and Positivity: Forgetting that applying an even power to an expression results in a positive value (e.g., \((x-1)^2 > 0\)).
• Square Root of a Square: Incorrectly simplifying \(\sqrt{x^2}\) as \(x\) instead of \(|x|\).
• Exponential Function and Inequalities: Misunderstanding how the increasing nature of \(y=e^x\) affects inequalities: if \(A < B\), then \(e^A < e^B\).
• Sign of \(e^N\) when \(N \ll 0\): Thinking \(e^N\) is large or negative when \(N \ll 0\). For \(N \ll 0\), \(e^N\) is a very small positive number.
• Absolute Value of \(e^{-x}\): Forgetting that \(|e^{-x}| = e^{-x}\) because \(e^{-x} > 0\).
• Sign of \(M = -\ln(\epsilon)\): Thinking \(M = -\ln(\epsilon)\) is negative when \(\epsilon\) is a very small positive number. In this case, \(\ln(\epsilon)\) is negative, so \(M = -\ln(\epsilon)\) is positive.
• Absolute Value of \(\frac{1}{x^p}\) for \(x>0\): Forgetting that if \(x>0\), then \(|\frac{1}{x^p}| = \frac{1}{x^p}\).