Fundamental Limit Laws in Calculus

Overview

This lecture introduces the fundamental limit laws used in calculus, demonstrates their application through various examples, and discusses techniques for evaluating limits when direct substitution fails.

Basic Limit Laws

• The limit of a constant as X approaches a is the constant itself.
• The limit of X as X approaches a is a.
• The sum and difference laws let you split limits over addition and subtraction.
• The constant multiple law allows you to factor constants out of limits.
• The product law applies the limit to each function in a product.
• The quotient law applies the limit to numerator and denominator, as long as the denominator's limit isn't zero.

Derived Limit Laws

• The power law: the limit of f(x)ⁿ is the limit of f(x) raised to n.
• The xⁿ law: the limit of xⁿ as X approaches a is aⁿ.
• The root law: the limit of the nth root of f(x) is the nth root of the limit, if n is even and the limit is ≥ 0.
• The nth root of x law: the limit of the nth root of x as X approaches a is the nth root of a.

Direct Substitution and Domain Considerations

• Direct substitution can be used if the value you're approaching is in the function's domain.
• For rational functions, ensure the denominator doesn’t become zero at the limit point.

Strategies When Direct Substitution Fails

• Factor and cancel common terms to remove discontinuities and use substitution.
• Rationalize the numerator or denominator if roots are involved.
• Expand and simplify expressions to reveal factors that can be canceled.
• Use sign analysis for rational functions near vertical asymptotes.
• Use one-sided limits for piecewise functions to determine if the overall limit exists.

Squeeze Theorem

• If a function is squeezed between two others with the same limit at a point, its limit at that point is the same.

Key Terms & Definitions

• Limit — The value a function approaches as the input approaches a certain point.
• Direct Substitution — Plugging the value into the function when it's within the domain.
• Removable Discontinuity — A point where a function is undefined but the limit exists.
• Squeeze Theorem — A method to find limits by bounding a function between two convergent functions.

Action Items / Next Steps

• Review and practice problems involving each limit law.
• Attempt problems where direct substitution fails and apply alternate strategies.
• Read textbook section on The Squeeze Theorem and its proofs.