Overview
This lecture introduces the concept of limits, showing how to evaluate them analytically (algebraic techniques) and graphically, including one-sided and two-sided limits, and types of discontinuities.
Introduction to Limits
- A limit describes the value a function approaches as the input (x) gets close to a specific number.
- Direct substitution tries to plug x into the function; if it results in 0/0 or undefined, other techniques are needed.
Analytical Methods for Evaluating Limits
- Substitute values close to, but not exactly at, the target x value to estimate the limit numerically.
- Factor expressions to cancel terms causing 0 in the denominator and then use direct substitution.
- For difference of cubes: factor x³–27 as (x–3)(x²+3x+9).
- For complex fractions, multiply by the common denominator to simplify.
- For limits with square roots, multiply numerator and denominator by the conjugate to eliminate the radical.
- If you end with a removable factor after simplification, cancel it, then substitute the limit value.
Evaluating Limits Graphically
- Identify the y-value the function approaches as x gets near the target from the left (−) or right (+).
- The left-sided limit and right-sided limit may differ; if not equal, the two-sided limit does not exist.
- The function value at a point is the filled (closed) circle on the graph, not necessarily the limit.
Types of Discontinuities
- Removable discontinuity (hole): point where limit exists but function is not defined, often due to a canceled factor.
- Jump discontinuity: graph "jumps" to a different value; one-sided limits differ.
- Infinite discontinuity: vertical asymptote; function increases or decreases without bound.
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a certain point.
- Direct Substitution — Plugging the input value directly into the function.
- Removable Discontinuity — A hole in the graph; discontinuity that can be fixed by redefining a single point.
- Jump Discontinuity — The function jumps from one value to another; limits from each side are different.
- Infinite Discontinuity — A vertical asymptote; function increases/decreases without bound near a point.
- Conjugate — An expression formed by changing the sign between two terms, used to simplify radicals.
Action Items / Next Steps
- Practice evaluating limits using substitution, factoring, and conjugate techniques.
- Try evaluating limits from given graphs, identifying one-sided and two-sided limits and types of discontinuity.
- Review and memorize formulas for factoring quadratics and cubes.