Overview
This lecture covers how to use limit notation to describe the behavior of rational functions near vertical asymptotes and holes, focusing on left-hand and right-hand limits.
Discontinuities in Rational Functions
- A value ( x = a ) is excluded from the domain if it makes the denominator zero, resulting in a discontinuity.
- Discontinuities can be vertical asymptotes (the graph goes to infinity) or holes (a gap at a point on the graph).
Understanding Limits Near Discontinuities
- Limits allow us to describe the function’s behavior as ( x ) approaches a discontinuity.
- The notation ( \lim_{x \to a^-} f(x) ) means approaching ( a ) from the left; ( \lim_{x \to a^+} f(x) ) means approaching from the right.
- At a vertical asymptote, the function values approach ( +\infty ) or ( -\infty ) from either side.
- At a hole, the limit from both sides is the ( y )-value where the hole occurs.
Examples and Techniques
- For vertical asymptotes, evaluate from both sides to determine if the function approaches ( +\infty ) or ( -\infty ).
- For holes, cancel common factors, substitute the ( x )-value of the hole into the simplified function, and this gives the y-coordinate of the hole.
- If only given the graph, look to see how the function behaves as it approaches the discontinuity.
- Without a graph, factor and analyze the function to find vertical asymptotes and holes, then use sign analysis near those values.
Step-by-Step Process
- Factor numerator and denominator to identify possible discontinuities.
- Shared factors (same multiplicity) indicate a hole; unique to denominator means a vertical asymptote.
- Use limits to describe behavior: for asymptotes, specify ( +\infty ) or ( -\infty ); for holes, give the hole’s y-value.
Key Terms & Definitions
- Discontinuity — a break or gap in the graph of a function, often where the denominator is zero.
- Vertical Asymptote — a vertical line ( x = a ) where the function increases or decreases without bound.
- Hole — a single point missing from the graph, found where numerator and denominator share a factor.
- Limit — describes the value a function approaches as ( x ) gets close to a particular point.
- Left-hand/Right-hand Limit — limit as ( x ) approaches from the left (( ^- )) or right (( ^+ )).
Action Items / Next Steps
- Practice identifying and classifying discontinuities in various rational functions.
- Practice calculating limits at vertical asymptotes and holes, with and without graphs.
- Review the method for finding the location of holes by simplifying rational functions.