Limits at Infinity
Analyzing horizontal asymptotes through the formal limit definition as x → ∞.
Introduction
What happens to as grows without bound?Limits at infinity describe the end behavior of functions and give us thehorizontal asymptotes we studied earlier.
Prerequisite Connection
You understand limits and horizontal asymptotes from graphing.
Today's Increment
We formalize end behavior using limit notation and rules.
Why This Matters
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
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