1. AP Calculus Chapter 1, Section 5
Infinite Limits

2. Graph the function 𝑓 𝑥 = 3 𝑥−2
Evaluate the limit from the left and the right as the function approaches 2.

3. A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite limit.
Just because you see the equal sign in lim f(x)=∞ doesn’t mean the limit exists. As you learned, the limit fails to exist since there is unbounded behavior.

4. Graph the equation. Find the value of c that does not exist in the domain. Then find the limit as the function approaches c from the left and right.
𝑓 𝑥 = 3 𝑥−4

5. 𝑓 𝑥 = 1 2−𝑥

6. 𝑓 𝑥 = 2 (𝑥−3) 2

7. 𝑓 𝑥 = −3 (𝑥+2) 2

8. Vertical Asymptotes
Vertical asymptotes occur when the denominator is equal to 0 (and the numerator is not 0).
Function will never touch an asymptote, but will forever become arbitrarily close.

9. Find the vertical asymptotes of the given functions.
𝑓 𝑥 = 1 2(𝑥+1)
𝑓 𝑥 = 𝑥 𝑥 2 −1
𝑓 𝑥 = cot 𝑥

10. Determine all vertical asymptotes and discuss the limit of the function as it approaches the VA from the left and right
𝑓 𝑥 = 𝑥 2 +2𝑥−8 𝑥 2 −4

11. Properties of Infinite Limits
Let c and L be real numbers and let f and g be functions such that
lim 𝑥→𝑐 𝑓(𝑥) =∞ and lim 𝑥→𝑐 𝑔 𝑥 =𝐿
Sum or difference: lim 𝑥→𝑐 𝑓 𝑥 ±𝑔 𝑥 =∞
Product: lim 𝑥→𝑐 𝑓 𝑥 𝑔 𝑥 =∞ , 𝐿>0
lim 𝑥→𝑐 𝑓 𝑥 𝑔 𝑥 =−∞ , 𝐿<0
Quotient: lim 𝑥→𝑐 𝑔(𝑥) 𝑓(𝑥) =0
Similar properties hold for one-sided limits and for functions for which the limit f(x) as x approaches c is -∞.

12. Because lim 𝑥→0 1=1 and lim 𝑥→0 𝑥 2 =∞ , you can write lim 𝑥→0 1+ 1 𝑥 2 =∞

13. Because lim 𝑥→ 1 − 𝑥 2 +1 =2 and lim 𝑥→ 1 − ( cot 𝜋𝑥)=−∞ , you can write lim 𝑥→ 1 − 𝑥 2 +1 cot 𝜋𝑥 =0

14. Because lim 𝑥→ 0 + 3=3 and lim 𝑥→ 0 + cot 𝑥 =∞, you can write lim 𝑥→ 0 + 3 cot 𝑥=∞

15. Before you leave class, page 90 #65

16. Homework
Pg. 88 – 90: #’s 1 – 49 every other odd, 59, 67,