Ch 11.2: Techniques for Evaluating Limits

Dividing Out Technique Used when direct substitution gives you a zero in the numerator and denominator Steps: 1)Factor numerator and denominator 2)Simplify the fraction by removing common factors 3)Use direct substitution on the portion left

Rationalizing Technique Used when direct substitution creates a zero in the numerator and denominator and there is a radical in the numerator Steps 1) Multiply numerator and denominator by conjugate 2) Simplify each portion, remove common factors 3) Find the limit of the remaining portion

One-sided Limits Used for limits that DNE because it approaches two different values from each side Steps: Give two answers 1)Find limit from left 2) Find limit from right

Graphing Calculator 3 ways can help you figure out a limit 1) Value: (TI-83)-Hit 2 nd, Trace, Value (TI-89)- Hit F5, Value 2) When value doesn’t work, use the table a) Set the table: (TI-83)-2 nd, Window (TI-89)- ♦, F4 b) View the table: (TI-83)-2 nd, Graph (TI-89)- ♦, F5 3) When a table doesn’t work, use TRACE

Remember, iff both the left and right limit exist and are equal to L From Left From Right Limits agree! Thus,

Calculus Example 1.Try direct substitution -Doesn’t work 2. Try Dividing Out -Factor out h -Simplify fraction -Solve new limit!