Overview
This lecture introduces two-sided infinite limits, provides their definitions and interpretations, and demonstrates examples using graphs and vertical asymptotes.
Two-Sided Infinite Limits: Definition & Interpretation
- A two-sided infinite limit describes behavior as x approaches a value a and f(x) increases/decreases without bound from both sides.
- If f(x) approaches positive infinity as x approaches a from both sides, we write: limₓ→ₐ f(x) = ∞.
- If f(x) approaches negative infinity as x approaches a from both sides, we write: limₓ→ₐ f(x) = −∞.
- The plus-minus (±) notation is often omitted, assuming both sides behave the same.
Determining Limit Existence
- If the left-sided and right-sided limits match and are both infinite (same sign), the two-sided infinite limit exists.
- If the behaviors differ or the function is not defined on one side, the two-sided infinite limit does not exist (DNE or TNE).
Example Analysis Using Graphs
- At x = −7, f(x) approaches negative infinity from both sides: limₓ→−7 f(x) = −∞.
- At x = −3, f(x) approaches positive infinity from both sides: limₓ→−3 f(x) = ∞.
- At x = 0, f(x) approaches positive infinity from both sides: limₓ→0 f(x) = ∞.
- At x = 6, f(x) approaches negative infinity from the left and positive infinity from the right, so the two-sided limit does not exist.
Key Terms & Definitions
- Two-sided infinite limit — Describes the behavior of a function approaching positive or negative infinity from both sides as x approaches a value.
- Vertical asymptote — A vertical line x = a where a function increases or decreases without bound as x approaches a.
Action Items / Next Steps
- Review class examples of two-sided infinite limits and practice identifying them using function graphs.