1.5.1: Resources and Key Concepts

Resources

Videos

• Finite limits at infinity
  • Limits at Infinity - Evaluating Finite Limits at Infinity

Key Concepts

Definitions

• Limits at Infinity (Informal):
  • Finite Limit at Infinity: If the values of \(f(x)\) become arbitrarily close to the finite value \(L\) as \(x\) becomes sufficiently large, we say the function \(f\) has a finite limit at infinity and write \( \displaystyle \lim_{x \to \infty} f(x) = L\).
  • Finite Limit at Negative Infinity: If the values of \(f(x)\) become arbitrarily close to the finite value \(L\) for \(x < 0\) as \(|x|\) becomes sufficiently large, we say the function \(f\) has a finite limit at negative infinity and write \( \displaystyle \lim_{x \to -\infty} f(x) = L\).
• Horizontal Asymptote: If \( \displaystyle \lim_{x \to \infty} f(x) = L\) or \( \displaystyle \lim_{x \to -\infty} f(x) = L\), where \(L\) is a finite number, we say the line \(y=L\) is a horizontal asymptote of \(f\).
• Infinite Limit at Infinity (Informal):
  • We say a function \(f\) has an infinite limit at infinity and write \( \displaystyle \lim_{x \to \infty} f(x) = \infty\) if \(f(x)\) becomes arbitrarily large for \(x\) sufficiently large.
  • We say a function \(f\) has a negative infinite limit at infinity and write \( \displaystyle \lim_{x \to \infty} f(x) = -\infty\) if \(f(x) < 0\) and \(|f(x)|\) becomes arbitrarily large for \(x\) sufficiently large. (Similar definitions apply for \(x \to -\infty\)).

Theorems

• Limit Laws for Limits at Infinity: Let \(f(x)\) and \(g(x)\) be defined for all \(x > a\), where \(a\) is a real number. Assume that \(L\) and \(M\) are real numbers such that \( \displaystyle \lim_{x \to \infty} f(x) = L\) and \( \displaystyle \lim_{x \to \infty} g(x) = M\). Let \(c\) be a constant. Then the Sum, Difference, Constant Multiple, Product, Quotient (for \(M \neq 0\)), Power, and Root Laws hold for limits at infinity. (These can be adjusted for \(x \to -\infty\)).
• Theorem (Limits of Powers of \(1/x\)): If \(p > 0\) is a rational number, then \( \displaystyle \lim_{x \to \infty} \frac{1}{x^p} = 0\). If \(p > 0\) is a rational number such that \(x^p\) is defined for all \(x\), then \( \displaystyle \lim_{x \to -\infty} \frac{1}{x^p} = 0\).
• Theorem (Common Limits at Infinity for Inverse Tangent and Hyperbolic Tangent):
  • \( \displaystyle \lim_{x \to \infty} \tan^{-1}(x) = \frac{\pi}{2}\)
  • \( \displaystyle \lim_{x \to -\infty} \tan^{-1}(x) = -\frac{\pi}{2}\)
  • \( \displaystyle \lim_{x \to \infty} b^{-x} = 0\) where \(b > 1\).
• Theorem: The Squeeze Theorem for Limits at Infinity:
  • Let \(f(x), g(x)\), and \(h(x)\) be defined for all \(x > a\). If \(f(x) \le g(x) \le h(x)\) for all \(x > a\) and \( \displaystyle \lim_{x \to \infty} f(x) = L = \lim_{x \to \infty} h(x)\), then \( \displaystyle \lim_{x \to \infty} g(x) = L\).
  • (A similar statement holds for \(x \to -\infty\)).
• Theorem (Common Infinite Limits at Infinity):
  • \( \displaystyle \lim_{x \to \infty} x = \infty\)
  • \( \displaystyle \lim_{x \to -\infty} x = -\infty\)
  • If \(p > 0\), \( \displaystyle \lim_{x \to \infty} x^p = \infty\)
  • If \(b > 1\), \( \displaystyle \lim_{x \to \infty} b^x = \infty\)

Common Mistakes

• Assuming a function cannot cross its horizontal asymptote: While a function cannot intersect a vertical asymptote, it can cross a horizontal asymptote, even an unlimited number of times (as shown with \(f(x) = \frac{\cos x}{x} + 1\) in Figure 3).
• Incorrectly handling \(\sqrt{x^2}\) when \(x \to -\infty\): Forgetting that \(\sqrt{x^2} = |x|\), and thus if \(x < 0\) (as when \(x \to -\infty\)), then \(\sqrt{x^2} = -x\). This is crucial in Example 2c.
• Misinterpreting indeterminate forms like \(\infty - \infty\): Assuming that \(\infty - \infty = 0\). This is an indeterminate form, and algebraic manipulation (like using conjugates in Example 2d) is required.