Overview
This lecture introduces the concept of limits in calculus, methods to evaluate them algebraically and graphically, and discusses different types of discontinuities in functions.
Introduction to Limits
- The limit asks what value f(x) approaches as x approaches a specific value.
- Direct substitution is the first method to try; if it gives an undefined value, try values close to the point.
- If direct substitution is undefined (e.g., 0/0), use algebraic simplification.
Algebraic Evaluation of Limits
- Factor numerators and denominators to cancel common terms, then use substitution.
- For complex fractions, multiply numerator and denominator by the least common denominator.
- For limits involving square roots, multiply by the conjugate to simplify.
Analytical and Numerical Examples
- Substitute values close to the point of interest to estimate limits if algebraic methods are unclear.
- Always check if direct substitution works before applying other methods.
Special Cases and Methods
- For one-sided limits, examine behavior as x approaches from either left or right.
- If left- and right-sided limits differ, the two-sided limit does not exist.
- Multiply by conjugates or common denominators for complex expressions.
Graphical Evaluation of Limits
- The left-hand limit is the value approached as x comes from the left.
- The right-hand limit is found as x comes from the right.
- If both one-sided limits are equal, the two-sided limit exists.
- Compare limits and function values at points to determine continuity.
Discontinuities
- Jump discontinuity (non-removable): Left and right limits are different.
- Removable discontinuity (hole): Limit exists but differs from the function value.
- Infinite discontinuity: Function approaches positive or negative infinity due to a vertical asymptote.
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a specified value.
- Direct substitution — Plugging the target x value directly into the function.
- Conjugate — An expression used to rationalize square root fractions, changing the sign between terms.
- Jump discontinuity — A break in the graph where left and right limits differ.
- Removable discontinuity — A single point gap (hole) in the graph where the limit exists but does not match the function value.
- Infinite discontinuity — A vertical asymptote where function grows without bound.
Action Items / Next Steps
- Practice evaluating limits using substitution, factoring, and conjugates.
- Review types of discontinuities and their graphical features.
- Complete assigned practice problems on limit evaluation and graph analysis.