Overview
This lecture introduces the concept of limits in calculus, explores analytical and graphical evaluation methods, and explains different types of discontinuities.
Introduction to Limits
- A limit asks what value a function (f(x)) approaches as x approaches a specific value.
- Direct substitution: Plugging in the value for x; if result is undefined (e.g., 0/0), alternate methods are needed.
- When direct substitution is undefined, evaluate the limit by plugging in numbers close to the target x-value.
Analytical Methods for Evaluating Limits
- Factorization: Simplify expressions and cancel terms to remove undefined values before substitution.
- For complex fractions, multiply numerator and denominator by the least common denominator.
- For functions with square roots, multiply by the conjugate to simplify the expression.
Example Problems
- Direct substitution works if no division by zero occurs (e.g., (\lim_{x \to 3} (x^2 + 5x - 4) = 20)).
- Factoring and canceling terms enables evaluation when substitution gives 0/0 (e.g., (\lim_{x \to 3} \frac{x^2 - 8x + 15}{x-3} = -2)).
- For complex fractions, manipulate algebraically to eliminate indeterminate forms (e.g., (\lim_{x \to 4} \frac{1/x - 1/4}{x-4} = -1/16)).
- Use the conjugate for limits involving roots (e.g., (\lim_{x \to 9} \frac{\sqrt{x}-3}{x-9} = 1/6)).
Evaluating Limits Graphically
- One-sided limits: Analyze the y-value as x approaches from the left or right.
- The limit exists only if both sides approach the same value.
- The function value at a point may differ from the limit.
Types of Discontinuities
- Jump Discontinuity: The left and right limits differ; limit does not exist.
- Removable Discontinuity: Limit exists but does not equal the function’s value (a "hole").
- Infinite Discontinuity: Function heads to infinity; typical at vertical asymptotes.
Example Discontinuity Problems
- For jump discontinuity, the left and right limits at a point differ (limit does not exist).
- For removable discontinuity, the limit exists but is not equal to f(a).
- For infinite discontinuity, function approaches infinity from one or both sides (limit does not exist).
Continuity
- A function is continuous at (x = a) if left limit, right limit, function value, and two-sided limit all exist and are equal.
Key Terms & Definitions
- Limit — The value a function approaches as x nears a specific point.
- Indeterminate Form — An expression like 0/0 where limit is not immediately clear.
- Direct Substitution — Plugging in the value of x directly into the function.
- Conjugate — A binomial formed by changing the sign between two terms, used for rationalizing.
- Jump Discontinuity — Discontinuity where left and right limits differ.
- Removable Discontinuity — Discontinuity where limit exists but does not match function value.
- Infinite Discontinuity — Discontinuity where function diverges to infinity.
Action Items / Next Steps
- Practice evaluating limits analytically and graphically.
- Review homework problems on limit evaluation (substitution, factoring, conjugates, and graph reading).
- Study types of discontinuities and their graphical representations.