Limits and Discontinuities Overview

Overview

This lecture introduces the concept of limits, explains how to evaluate them analytically and graphically, and covers common types of discontinuities.

Introduction to Limits

• Limits help determine the value a function approaches as the input approaches a specific point.
• If direct substitution gives 0/0 or undefined, other methods must be used to evaluate the limit.

Evaluating Limits Analytically

• For simple functions, use direct substitution to find the limit.
• If substitution gives 0/0, try factoring to simplify and cancel terms before substituting.
• For complex fractions, multiply numerator and denominator by the common denominator to simplify.
• For expressions with square roots, multiply by the conjugate to eliminate the root and simplify.

Worked Examples

• Factoring quadratics or cubes can resolve 0/0 forms for limits like (x²–4)/(x–2) and (x³–27)/(x–3).
• For complex fractions like [1/x – 1/3]/(x–3), use common denominators and simplification.
• When a square root is present, like [√x–3]/(x–9), use the conjugate to rationalize and cancel terms.

Evaluating Limits Graphically

• The limit from the left (left-sided limit) and right (right-sided limit) may be different.
• If the left and right limits are equal, the two-sided limit exists and equals that value.
• If they differ, the two-sided limit does not exist.
• The function value f(a) is found by the filled (closed) circle at x=a on the graph.
• Vertical asymptotes indicate points where the limit approaches infinity or does not exist.

Types of Discontinuities

• Removable discontinuity (hole): the graph has a "hole" at a point but can be "fixed".
• Jump discontinuity: the graph jumps from one value to another (not removable).
• Infinite discontinuity: the graph goes to infinity (vertical asymptote, not removable).

Key Terms & Definitions

• Limit — The value a function approaches as the input approaches some value.
• Direct Substitution — Plugging the approaching value directly into the function.
• Factoring — Rewriting an expression to reveal factors that may cancel problematic terms.
• Conjugate — A binomial formed by changing the sign in a sum or difference with a square root.
• One-sided Limit — Limit taken from either the left or right side of a point.
• Removable Discontinuity — A "hole" in the graph where the limit exists but the function is undefined.
• Jump Discontinuity — A sudden change in function value, limit does not exist.
• Infinite Discontinuity — The function increases or decreases without bound near a point (vertical asymptote).

Action Items / Next Steps

• Practice evaluating limits analytically using factoring, conjugates, and complex fraction methods.
• Review graphical examples to identify and classify discontinuities.
• Complete homework problems on evaluating limits and identifying discontinuity types.