Calculus Chapter 2: Limits Section 2.7: Limits at Infinity SWBAT: evaluate trig limits as the function approaches infinity. Standard: F.IF. Ø Understand the concept of a function and the use of function notation. “Do Now” Given the graph of 𝑓(𝑥), find each of the indicated limits. a) lim!→!! 𝑓 𝑥 = b) lim!→!! 𝑓 𝑥 = c) lim!→!! 𝑓 𝑥 = d) lim!→!! 𝑓 𝑥 = e) lim!→! 𝑓 𝑥 = f) lim!→! 𝑓 𝑥 = • To date, we’ve discussed limits at particular x-­‐values (for example as x approaches 3). However, some functions will have limits at infinity. Some notation before we begin: Ø lim!→! 𝑓 𝑥 = 𝐿 if 𝑓(𝑥) gets closer and closer to L as 𝑥 → ∞. Ø lim!→!! 𝑓 𝑥 = 𝐿 if 𝑓(𝑥) gets closer and closer to L as 𝑥 → −∞. What will be the effect on the graph? Calculus Chapter 2: Limits Why? 100! = 100!! = 10,000! = 1000!! = 10,000!! = (−100)! = (−100)!! = (−1000)! = (−1000)!! = (−10,000)! = (−10,000)!! = Why? (−100)! = (−1000)! = (−10,000)! = 1000! = Calculus Chapter 2: Limits Example: Calculate 20𝑥 ! − 3𝑥
lim
!→! 3𝑥 ! − 4𝑥 ! + 5
Does this limit have and indeterminate form? Can we simplify using one of our four methods? So what do we do? First, we divide both the numerator and denominator by the highest power of x in the denominator. Second, “Plug in” infinity and evaluate. Example: Calculate 9𝑥 ! − 2
lim
!→! 6 − 29𝑥
First, we divide both the numerator and denominator by the highest power of x in the denominator. Second, “Plug in” infinity and evaluate. Calculus Chapter 2: Limits Example: Calculate lim
!→!
3𝑥
!
!
+ 7𝑥
𝑥! − 𝑥
!
!!
!
!
First, we divide both the numerator and denominator by the highest power of x in the denominator. Second, “Plug in” infinity and evaluate. Practice Problems: a) lim!→!
!! ! !!"!
!! ! !!
b) lim!→!
!
!!!
c) lim!→!!
!!!!
!! ! !!!!!
Closure: We’ve talked about limits at particular x-­‐values. Today, we talked about limits as x goes to infinity. If the limit of a function as x goes to positive/negative infinity is a particular number, what will be the effect on the graph? Homework: Pg. 105 #’s 1 – 5, 7 – 15 Odd Calculus Chapter 2: Limits “Do Now” Find the limits at infinity. Yesterday, we learned about evaluating the limit of a function as x approaches positive/negative infinity. To do this, we divided each term by the highest power of x in the denominator. This process always yields 1 of 3 cases: Case 1: If the degree of x in the denominator is higher, the limit is 0. From the other day: 20𝑥 ! − 3𝑥
lim
!→! 3𝑥 ! − 4𝑥 ! + 5
Case 2: If the degree of x in the numerate is higher, the limit is either positive or negative infinity. From the other day: 9𝑥 ! − 2
lim
!→! 6 − 29𝑥
Case 3: If the degree of x in the numerator = the degree of x in the denominator, the limit is the ratio of the leading coefficients. From the other day: 3𝑥 ! + 20𝑥
lim
!→! 4𝑥 ! + 9
Example: Evaluate each limit a) lim!→!!
c) lim!→!!
!!!!
!! ! !!!!!
!! ! !!! ! !!"!
!! ! !!"! ! !!"
b) lim!→!
!! ! !!!!
!! ! !!
Calculus Chapter 2: Limits Although, there is one slight problem to this process… For 𝑥 > 0, 12𝑥 + 25
lim
!→! 16𝑥 ! + 100𝑥 + 500
However, if 𝑥 < 0, 𝑡ℎ𝑒𝑛 𝑥 = − 𝑥 ! 12𝑥 + 25
lim
!→!! 16𝑥 ! + 100𝑥 + 500
Will we have a problem with this example? !
8𝑥 ! + 7𝑥 !
lim
!→!!
16𝑥 ! + 6
How about this example? 𝑥+1
lim
!→!! 4𝑥 ! + 1
v Finding limits of functions as they approach infinity can help us find the horizontal asymptotes of a given function. Find the horizontal asymptote(s). Verify using the calculator. a) 𝑓 𝑥 =
c) 𝑓 𝑥 =
!! ! !!!!!
!! ! !!
!"! ! !!
!! ! !!
b) 𝑓 𝑥 =
!"! ! !!
!!!!
Calculus Chapter 2: Limits Closure: 1) What are the three main cases for evaluating the limit of a function as x approaches infinity? 2) How can we use limits to find horizontal asymptotes? Homework: Pg. 105 #’s 17 – 19 and 23 – 26