Overview
This lecture focused on using analytic (algebraic) methods to find limits, including evaluating limits at points of discontinuity, understanding the behavior near holes and vertical asymptotes, and applying special techniques such as factoring, the conjugate method, and handling piecewise functions.
Connecting Multiple Representations of Limits
- A function can have a limit at a point where it is undefined (such as a hole).
- The limit at a hole is determined by the approaching values of the graph from both sides.
- At vertical asymptotes, one-sided limits approach either positive or negative infinity.
- For a vertical asymptote, possible two-sided limits are: both sides go to positive infinity, both to negative infinity, or don't exist if they diverge in opposite directions.
- Direct substitution is always tried first; 0/0 or nonzero/0 forms signal the need for further analysis.
Algebraic Techniques and Discontinuities
- If direct substitution gives 0/0, algebraic manipulation (factoring, cancellation, etc.) is needed to find the limit.
- 0/0 indicates a hole (removable discontinuity); nonzero/0 often signals a vertical asymptote.
- Removable discontinuities (holes) occur where a factor cancels from numerator and denominator.
- To find the sign of infinity at an asymptote, analyze the sign of the result when approaching from left/right of the asymptote.
- For rational functions, if after simplification and substitution you get a number, it’s a hole; if you get ±infinity, it’s a vertical asymptote.
Limits of Piecewise Functions
- For piecewise functions, use the correct "piece" based on which interval the approaching value falls into.
- Evaluate one-sided limits by plugging values just less or just greater than the target point into the appropriate function piece.
- If both one-sided limits are equal, the two-sided limit exists and equals that value (Limit Existence Theorem).
- If left- and right-hand limits differ, a jump discontinuity exists.
Complex Fractions and Conjugate Method
- For limits involving complex fractions, eliminate denominators using the least common multiple (LCM) method before substituting.
- For limits with radicals (square roots) leading to 0/0, multiply by the conjugate to eliminate the radical.
- Only multiply out the part containing the radical when using the conjugate.
- After simplifying, substitute to find the limit.
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a certain point.
- Hole/Removable Discontinuity — A point where a function is undefined but the limit exists, usually due to a factor that cancels in numerator and denominator.
- Vertical Asymptote — A value of x where the function increases/decreases without bound (±infinity).
- Jump Discontinuity — A sudden "jump" or break in the graph, often caused by piecewise functions.
- Conjugate — For an expression a+b, the conjugate is a–b; used to rationalize radicals in limits.
- One-sided Limit — The value as x approaches from only one direction (left or right).
- Complex Fraction — A fraction where the numerator, denominator, or both contain fractions.
- LCM Method — Multiplying numerator and denominator by the least common denominator to simplify complex fractions.
Action Items / Next Steps
- Review factoring techniques and practice canceling common factors.
- Complete any assigned worksheets or problems on analytic limit methods.
- Watch posted videos for additional examples, especially on factoring and synthetic division.
- Prepare for upcoming quiz on limits, emphasizing graphs and algebraic methods.