Limits at Infinity
Analyzing horizontal asymptotes through the formal limit definition as x → ∞.
Introduction
Analyzing horizontal asymptotes through the formal limit definition as x → ∞.
Past Knowledge
You understand limits and horizontal asymptotes from graphing.
Today's Goal
We formalize end behavior using limit notation and rules.
Future Success
End behavior predicts long-term trends in growth models and physical systems.
Key Concepts
Basic Limits at Infinity
Rational Function Rule
Compare highest powers in numerator and denominator:
- • Numerator degree < denominator → limit is 0
- • Degrees equal → limit is ratio of leading coefficients
- • Numerator degree > denominator → limit is ±∞
Divide by Highest Power
For complex expressions, divide every term by the highest power of x in the denominator.
Worked Examples
Example 1: Equal Degrees (Basic)
Find
Degrees equal (both 2), so limit = ratio of leading coefficients:
Limit = 3/5
Example 2: Numerator Smaller (Intermediate)
Find
Divide all terms by :
Limit = 0
Example 3: With Radicals (Advanced)
Find
Since , :
Common Pitfalls
Forgetting the sign at -∞
As , . Watch for sign changes!
Saying ∞/∞ = 1
∞/∞ is indeterminate. You must simplify first.
Real-World Application
Long-Term Population Growth
Carrying capacity models use limits at infinity to predict maximum sustainable populations:
where K is the carrying capacity.
Practice Quiz
Practice Quiz
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