1. LIMITS

2. If , find the following value of x in given table.

3. What is Limits? • A limit is the maximum or minimum value that a given function. • A limit will equal or approach (get closer to). • Example: a car can get faster and faster; however, at some point it will reach the maximum speed that it can possibly go. Therefore we say that the car has reached it’s maximum speed limit.

4. Back to previous table. The answer is: • What can you deduce about that function?

5. That means: Limit f(x) if x approach to 1, and if writes to determine the value of limit, we can choose x ε R that value of x apprroach whether form left or right of x. • So:

6. 3 Methods to Calculate Limit of Function • Table of Limit • Substitution if x = 1 substitute to , then f(x) = 3. This result is the same with value of that limit function. example: find of this following limit: a) b) c)

7. However, in several problems, we can not do substitution method. • Example: find the value of • If x = 2 substitute to , the answer is . • It means that the limit doesn’t have value. • The problems must calculate with different method.

8. Factorization example: find this following limits: a. b.

10. Determine the value of limit using rationalize • Simplify the square root of function. example: simplify ! answer:

11. one more example: simplify ! answer:

12. Determine limit function with rationalizing. • Find the value of : ! • Though we’ve already know the simple form of , then we can substitute x = 10 to

13. The answer:

14. Exercise: find the value of this following limit a. b. c.

15. Limit Function For x Approach to Infinity A. Form . example: find the value of limit ! answer : =

16. exercise: form the example, we can deduce that:

17. Limit Function For x Approach to Infinity B. Form . To find the value of this limit, we have to multiply the limit with , and the form will be: = =

18. from previous sentence, we can deduce that: example: find the value of this following limit. a. b.