Evaluating Limits Algebraically
Using substitution, factoring, and rationalization to resolve 0/0 indeterminates.
Introduction
Using substitution, factoring, and rationalization to resolve 0/0 indeterminates.
Past Knowledge
You understand limit notation and one-sided limits.
Today's Goal
We master algebraic techniques to resolve indeterminate forms.
Future Success
These techniques are essential for computing derivatives and solving real calculus problems.
Key Concepts
Method 1: Factoring
Factor numerator and denominator, cancel common factors, then substitute.
Method 2: Rationalizing
Multiply by conjugate to eliminate radicals:
Method 3: Simplifying Complex Fractions
Combine fractions in numerator/denominator, then simplify.
Special Limits
Worked Examples
Example 1: Factoring (Basic)
Find
Limit = 6
Example 2: Rationalizing (Intermediate)
Find
Multiply by conjugate:
Limit = 1/4
Example 3: Complex Fraction (Advanced)
Find
Combine fractions in numerator:
Limit = -1/4
Common Pitfalls
Canceling before factoring completely
Make sure you've fully factored both numerator and denominator.
Forgetting to multiply both top and bottom
When rationalizing, the conjugate goes in both numerator and denominator.
Real-World Application
Computing Derivatives
The derivative formula always gives 0/0 when h→0. These algebraic techniques are exactly what we use to evaluate derivatives!
Practice Quiz
Practice Quiz
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