1.3.1: Resources and Key Concepts

Resources

Videos

• Limit Laws
  • Limits - Limit Laws - Concepts, Theorems, and Proof
• Evaluating limits using the Limit Laws
  • Limits - Limit Laws - Evaluating Limits Using the Limit Laws
• Power Law for limits
  • Proofs in Differential Calculus: Limits of Powers
• Direct Substitution Property
  • Limits - Limit Laws - Direct Substitution: Theorem, Proof, and Example
• Squeeze Theorem
  • Limits - The Squeeze Theorem - Concept, Theorem, and Examples
  • Limits - The Squeeze Theorem - Concept, Theorem, and Examples: Extra Example
• Limit of sin(x) as x approaches 0
  • Proofs in Differential Calculus - \( \displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1 \)
  • Derivatives of Trigonometric Functions - Using the Lemmas to Evaluate Limits
• Limit of cos(x) as x approaches 0
  • Proofs in Differential Calculus - \( \displaystyle \lim_{x \to 0} \frac{\cos(x) - 1}{x} = 0 \)

Key Concepts

Theorems

• Corollary: Basic Limit Laws: For any real number \(a\) and any constant \(c\),
  1. \( \displaystyle \lim_{x \to a} x = a\)
  2. \( \displaystyle \lim_{x \to a} c = c\)
• Limit Laws: Let \(f(x)\) and \(g(x)\) be defined for all \(x \neq a\) over some open interval containing \(a\). Assume that \(L\) and \(M\) are real numbers such that \( \displaystyle \lim_{x \to a} f(x) = L\) and \( \displaystyle \lim_{x \to a} g(x) = M\). Let \(c\) be a constant. Then:
  • Sum Law for Limits: \( \displaystyle \lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = L + M\)
  • Difference Law for Limits: \( \displaystyle \lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) = L - M\)
  • Constant Multiple Law for Limits: \( \displaystyle \lim_{x \to a} cf(x) = c \cdot \lim_{x \to a} f(x) = cL\)
  • Product Law for Limits: \( \displaystyle \lim_{x \to a} (f(x) \cdot g(x)) = \left(\lim_{x \to a} f(x)\right) \cdot \left(\lim_{x \to a} g(x)\right) = L \cdot M\)
  • Quotient Law for Limits: \( \displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{L}{M}\), for \(M \neq 0\).
  • Power Law for Limits: \( \displaystyle \lim_{x \to a} (f(x))^n = \left(\lim_{x \to a} f(x)\right)^n = L^n\), for every positive integer \(n\).
  • Root Law for Limits: \( \displaystyle \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} = \sqrt[n]{L}\), for all \(L\) if \(n\) is odd and for \(L \ge 0\) if \(n\) is even.
• Theorem: Direct Substitution Property (for polynomial and rational functions): Let \(p(x)\) and \(q(x)\) be polynomial functions. Let \(a\) be a real number. Then, \( \displaystyle \lim_{x \to a} p(x) = p(a)\), and \( \displaystyle \lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}\) when \(q(a) \neq 0\).
• One-Sided Limits (application of Limit Laws): Simple modifications to the Limit Laws allow them to be applied to one-sided limits. For \( \displaystyle \lim_{x \to a^-} h(x)\), \(h(x)\) must be defined over an open interval of the form \((b,a)\). For \( \displaystyle \lim_{x \to a^+} h(x)\), \(h(x)\) must be defined over an open interval of the form \((a,c)\).
• Theorem: Squeeze Theorem: Let \(f(x)\), \(g(x)\), and \(h(x)\) be defined for all \(x \neq a\) over an open interval containing \(a\). If \(f(x) \le g(x) \le h(x)\) for all \(x \neq a\) in an open interval containing \(a\) and \( \displaystyle \lim_{x \to a} f(x) = L = \lim_{x \to a} h(x)\) where \(L\) is a real number, then \( \displaystyle \lim_{x \to a} g(x) = L\).
• Theorem: Limits Involving Sine and Cosine:
  • \( \displaystyle \lim_{\theta \to 0} (\sin(\theta)) = 0\)
  • \( \displaystyle \lim_{\theta \to 0} (\cos(\theta)) = 1\)
• Theorem: Important Trigonometric Limits:
  • \( \displaystyle \lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1\)
  • \( \displaystyle \lim_{\theta \to 0} \frac{1-\cos\theta}{\theta} = 0\)

Common Mistakes

• The Limit Laws Require Limits to Exist: Thinking that the Limit Laws apply to every limit encountered. To use the Limit Laws, both of the component limits (e.g., \( \displaystyle \lim_{x \to a} f(x)\) and \( \displaystyle \lim_{x \to a} g(x)\)) must exist and be finite.
• Misapplying the Direct Substitution Property: The Direct Substitution Property for rational functions \( \displaystyle \lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}\) can only be applied if the limit in the denominator \(q(a)\) is nonzero. An answer will only be "DNE" if both left- and right-hand limits have been evaluated and found not to be equal (or if one or both tend to infinity differently).
• Incorrectly Applying Limit Laws to Non-Finite Limits (Breaking the Limit Laws): Applying Limit Laws when one or more component limits are infinite. For example, if \( \displaystyle \lim_{x \to a} f(x) = 0\) and \( \displaystyle \lim_{x \to a} g(x) = \infty\), one cannot use the Product Law to conclude \( \displaystyle \lim_{x \to a} f(x)g(x) = 0 \cdot \infty\). Algebraic simplification is needed first.
• Using Degree Measure in Calculus for Trigonometric Functions: The proofs of trigonometric limits (e.g., \( \displaystyle \lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1\)) rely on radian measure. Calculus and degree measure do not get along for trigonometric functions.