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Section 1.6 Skills: Calculating Limits Using the Limit Laws

Section 1.6: Calculating Limits Using the Limit Laws

This section provides the tools for computing limits algebraically using limit laws and algebraic techniques.

Skills in This Section

Skill Description Difficulty
Limit Laws Fundamental rules for computing limits Beginner
Direct Substitution When and how to substitute directly Beginner
Algebraic Limit Techniques Factoring, rationalizing, simplifying Intermediate
One-Sided Limit Calculation Computing left and right limits Intermediate
Indeterminate Forms (Algebraic) Resolving $\frac{0}{0}$ cases Intermediate
Squeeze Theorem Bounding technique for tricky limits Intermediate

The Limit Laws

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then:

  1. Sum: $\lim_{x \to a} [f(x) + g(x)] = L + M$
  2. Difference: $\lim_{x \to a} [f(x) - g(x)] = L - M$
  3. Product: $\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M$
  4. Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$ (if $M \neq 0$)
  5. Power: $\lim_{x \to a} [f(x)]^n = L^n$

Learning Path

graph TD
    A["Limit Laws"] --> B["Direct Substitution"]
    B --> C["Algebraic Techniques"]
    C --> D["Indeterminate Forms"]
    A --> E["One-Sided Limits"]
    D --> F["Squeeze Theorem"]

    click A "limit-laws.html"
    click B "direct-substitution.html"
    click C "algebraic-limit-techniques.html"

    click D "indeterminate-forms-algebraic.html"
    click E "../ch1-sec5/one-sided-limits.html"
    click F "squeeze-theorem.html"

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