Limits and Discontinuities

Overview

This lecture introduces the concept of limits, methods for evaluating them analytically and graphically, and discusses different types of discontinuities in functions.

Introduction to Limits

• The limit of a function describes its behavior as x approaches a specific value.
• Direct substitution can be used if plugging in the value does not yield an undefined expression.

Analytical Evaluation of Limits

• If direct substitution gives 0/0 (indeterminate), use values close to the target to estimate the limit.
• For rational functions, factor and cancel terms to eliminate the zero in the denominator before substitution.
• Example: (\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4) after factoring and cancelling (x - 2).
• Use special factorization formulas, such as the difference of cubes, to simplify expressions.
• Example: (\lim_{x \to 3} \frac{x^3 - 27}{x - 3} = 27), after using the difference of cubes formula.

Limits Involving Complex Fractions and Radicals

• Multiply numerator and denominator by the common denominator to simplify complex fractions.
• Multiply by the conjugate when dealing with expressions containing square roots to rationalize.
• After simplification and cancellation, use direct substitution to find the limit.

Graphical Evaluation of Limits

• To evaluate a limit graphically, observe the y-value as x approaches from the left and right.
• If both one-sided limits match, the overall limit exists and equals that value.
• If they do not match, the limit does not exist at that point.

Discontinuities in Functions

• Hole (Removable discontinuity): The graph is missing a point, but the limit exists.
• Jump discontinuity: The graph jumps to a different value; limits from each side differ; non-removable.
• Infinite discontinuity (Vertical asymptote): The function approaches infinity or negative infinity; undefined at that point.

Key Terms & Definitions

• Limit — The value a function approaches as x approaches a specific number.
• Direct Substitution — Plugging the value directly into the function to find the limit.
• Indeterminate Form — An undefined result such as 0/0 when evaluating a limit.
• Conjugate — An expression with the same terms but opposite sign between them, used to simplify radicals.
• One-sided Limit — The value approached by the function as x approaches from only one side (left or right).
• Removable Discontinuity — A hole in the graph where the limit exists but the function is undefined.
• Non-removable Discontinuity — Either a jump or infinite discontinuity where the limit does not exist.

Action Items / Next Steps

• Practice evaluating limits analytically through factoring, conjugates, and direct substitution.
• Work through graphical limit problems for one-sided and two-sided limits.
• Identify types of discontinuities in various functions.