Warm up Seek and Solve!!!
Seek and Solve Answers: 0 2 DNE 3
Investigation # 1 Use the graph of y = 2 below to find the following limits: 1. lim x 2 2 = 3 2. lim x 0 2 = 3 3 3. lim x 3 2 = 3
Basic Limit Laws 1. lim x a k = k *Where k is a constant
Investigation # 2 Use the graph of y = x below to find the following 5 limits: 1. lim x 2 x = 2. lim x 0 x = 5 5 3. lim x 1 x = 5 4. lim x 4 x =
2. lim x a x = a Basic Limit Laws
Basic Limit Laws 3. 4.
Techniques for Solving Limits Algebraically #1 Technique Direct Substitution When You Use It We always try it 1st! Example
Additional Example of Technique #1
Your Turn: Use direct substitution to solve problems 1 4 on the Calculating Limits Practice handout. 1. 2. 3. 4.
Techniques for Solving Limits Algebraically #2 Technique Cancelling Common Factors When You Use It When you have a rational equation AND direct substitution yields Example
Additional Example of Technique #2
Your Turn: Use direct substitution to solve problems 5 10 on the Calculating Limits Practice handout. 5. 6.
7. 8.
9. 10.
Techniques for Solving Limits Algebraically #3 Technique Sign Analysis When You Use It When you have a rational equation AND the combination of direct substitution and/or cancelling common terms yields *You need to pay attention to whether you have a right hand limit, a left hand limit, or a two sided limit. Example
Additional Examples of Technique #3
Additional Examples of Technique #3 When you have a two sided limit and you're using sign analysis, then you need to evalute the limit from BOTH sides.
Your Turn: Use direct substitution to solve problems 11 16 on the Calculating Limits Practice handout. 11. 12.
13. 14.
15. 16.
Rationalizing Review!!! We can rationalize numerators and denominators! We can rationalize monomials and binomials; but, we're going to focus on binomials because we will encounter those the most.
Binomial Conjugates The same binomial that you want to rationalize, EXCEPT that the sign inbetween the two terms is the OPPOSITE of the original binomial.
Rationalizing Binomials Step 1: Identify the expression that you want to rationalize and its conjugate. Step 2: Multiply the fraction by Step 3: Multiply the two fractions. (You may need to FOIL!) Step 4: Simplify the fraction and reduce it to least radical form.
Example #1
Example #2
Example #3
Your Turn: Complete problems 1 4 on the Rationalizing Review handout.
1. 2.
3. 4.
Techniques for Solving Limits Algebraically #4 Technique Rationalizing the Numerator or the Denominator When You Use It When direct substitution doesn't work AND you have a radical in either the numerator or the denominator. Example
Additional Example Using Technique #4
Additional Example Using Technique #4
Your Turn: Complete problems 17 22 on the Calculating Limits Practice handout.
17.
18.
19.
20.
21.
22.
Choosing the Correct Technique for Calculating Limits Flow Chart
Your Turn: Complete problems 1 6 on the Calculating Limits Practice Part II handout.
1. 2.
3. 4.
5. 6.
Calculating the Limit of a Piecewise Defined Function You need to check the restricted domains to determine which expression to evaluate
If you're calculating the two sided limit at a point where the expression changes, you need to evaluate BOTH the right hand AND the left hand limits for that point. Remember, in order for the two sided limit to exist, the right hand limit MUST EQUAL the left hand limit.
Your Turn: Complete problems 7 14 on the Calculating Limits Practice Part II handout.
7. 8. 9. 10. 11. 12. 13. 14.