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Limits and Discontinuities

Aug 26, 2025

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.