Overview
This lecture explains how to solve one-sided limits graphically and algebraically, and covers how one-sided limits relate to the existence of a standard (two-sided) limit.
Regular vs. One-Sided Limits
- A regular (two-sided) limit approaches a value from both the left and right sides.
- A one-sided limit approaches a value from only one side—either left or right.
- A negative sign (e.g., (5^-)) indicates approaching from the left; a positive sign (e.g., (5^+)) indicates approaching from the right.
Graphical Approach to One-Sided Limits
- For (\lim_{x \to 5^-} \frac{3}{x-5}), as (x) approaches 5 from the left, (y) decreases without bound toward (-\infty).
- For (\lim_{x \to 5^+} \frac{3}{x-5}), as (x) approaches 5 from the right, (y) increases without bound toward (+\infty).
Algebraic Approach to One-Sided Limits
- To evaluate from the left, pick a value just less than 5 and substitute into the function.
- A value slightly less than 5 gives a small negative denominator, making the fraction large and negative, so the limit is (-\infty).
- To evaluate from the right, pick a value just greater than 5; this gives a small positive denominator, making the fraction large and positive, so the limit is (+\infty).
Determining Existence of Two-Sided Limit
- If the left- and right-sided limits are not equal, the two-sided (regular) limit does not exist.
- In this example, left limit is (-\infty) and right limit is (+\infty), so the limit as (x \to 5) does not exist.
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a certain point.
- One-Sided Limit — The value a function approaches as the input approaches from only one side (left or right).
- Graphical Approach — Using a graph to visualize function behavior near the point of interest.
- Algebraic Approach — Substituting numbers close to the target value to evaluate the limit numerically.
- Does Not Exist (DNE) — Used when a limit does not settle to a single value from both sides.
Action Items / Next Steps
- Review videos on introduction to limits and solving limits by factoring if not already done.
- Practice evaluating one-sided limits both graphically and algebraically.
- Prepare to justify when a limit does not exist using one-sided limits.