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Infinite Limits and Limits at Infinity

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1 Infinite Limits and Limits at Infinity
Section 1.5 and 3.5 Calculus BC AP/Dual, Revised Β©2017 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

2 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Infinite Limits are limits in which 𝒇(𝒙) increases or decreases without bound as 𝒙 approaches 𝒄 or π₯𝐒𝐦 𝒙→𝒄 𝒇(𝒙) =±∞, does not exist. The statement, β€œπ’™ approaches to infinity” really describes the limit to reach a constant number. Let 𝒇 and π’ˆ be continuous on an open interval containing 𝒄. If 𝒇(𝒄)β‰  𝟎, π’ˆ(𝒄) = 𝟎, then 𝒉 𝒙 = 𝒇 𝒙 π’ˆ 𝒙 has a vertical asymptote at 𝒙=𝒄. 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

3 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Steps Simplify the rational function Establish the limit from the left side and right side If the limits are the same, it is the established limit. If the limits are not the same, the limit fails to exist. 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

4 Review Determine π₯𝐒𝐦 𝒙→ 𝟐 βˆ’ πŸ‘ π’™βˆ’πŸ and π₯𝐒𝐦 𝒙→ 𝟐 + πŸ‘ π’™βˆ’πŸ 𝒙 𝟏.πŸ— 𝟏.πŸ—πŸ—
𝟏.πŸ—πŸ—πŸ— 𝟐 𝟐.𝟎𝟎𝟏 𝟐.𝟎𝟏 𝟐.𝟏 𝒇(𝒙) 𝒇(𝒙) βˆ’πŸ‘πŸŽ βˆ’πŸ‘πŸŽπŸŽ βˆ’πŸ‘πŸŽπŸŽπŸŽ πŸ‘πŸŽπŸŽπŸŽ πŸ‘πŸŽπŸŽ πŸ‘πŸŽ 𝒇(𝒙) βˆ’πŸ‘πŸŽ πŸ‘πŸŽ 𝒇(𝒙) β€“πŸ‘πŸŽ β€“πŸ‘πŸŽπŸŽ πŸ‘πŸŽπŸŽ πŸ‘πŸŽ 𝒇 𝒙 DECREASES without bound 𝒇 𝒙 INCREASES without bound 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

5 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Review Determine π₯𝐒𝐦 𝒙→ 𝟐 βˆ’ πŸ‘ π’™βˆ’πŸ and π₯𝐒𝐦 𝒙→ 𝟐 + πŸ‘ π’™βˆ’πŸ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

6 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 1 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟏 𝟐 π’™βˆ’πŸ and π₯𝐒𝐦 π’™β†’πŸ = 𝟏 𝟐 π’™βˆ’πŸ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

7 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 1 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟏 𝟐 π’™βˆ’πŸ and π₯𝐒𝐦 π’™β†’πŸ = 𝟏 𝟐 π’™βˆ’πŸ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

8 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 1 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟏 𝟐 π’™βˆ’πŸ and π₯𝐒𝐦 π’™β†’πŸ = 𝟏 𝟐 π’™βˆ’πŸ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

9 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 2 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟐 𝐬𝐒𝐧 𝒙 and π₯𝐒𝐦 π’™β†’πŸŽ = 𝟐 𝐬𝐒𝐧 𝒙 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

10 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 2 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟐 𝐬𝐒𝐧 𝒙 and π₯𝐒𝐦 π’™β†’πŸŽ = 𝟐 𝐬𝐒𝐧 𝒙 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

11 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 2 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟐 𝐬𝐒𝐧 𝒙 and π₯𝐒𝐦 π’™β†’πŸŽ = 𝟐 𝐬𝐒𝐧 𝒙 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

12 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Your Turn Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟏 𝒙 𝟐 and π₯𝐒𝐦 π’™β†’πŸŽ = 𝟏 𝒙 𝟐 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

13 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Infinite Limits are limits in which 𝒇(𝒙) increases or decreases without bound as 𝒙 approaches 𝒄 or π₯𝐒𝐦 𝒙→𝒄 𝒇(𝒙) =±∞, does not exist. The statement, β€œπ’™ approaches to infinity” really describes the limit to reach a constant number. Let 𝒇 and π’ˆ be continuous on an open interval containing 𝒄. If 𝒇(𝒄)β‰  𝟎, π’ˆ(𝒄) = 𝟎, then 𝒉 𝒙 = 𝒇 𝒙 π’ˆ 𝒙 has a vertical asymptote at 𝒙=𝒄. 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

14 Properties of Infinite Limits
As we let π₯𝐒𝐦 𝒙→𝒄 𝒇 𝒙 =∞ and π₯𝐒𝐦 𝒙→𝒄 π’ˆ 𝒙 =𝑳 : Types: Sum or difference: π₯𝐒𝐦 𝒙→𝒄 𝒇 𝒙 Β±π’ˆ 𝒙 =βˆžΒ±π‘³=∞ Product: π₯𝐒𝐦 𝒙→𝒄 𝒇 𝒙 π’ˆ 𝒙 =∞ if π₯𝐒𝐦 𝒙→𝒄 π’ˆ 𝒙 or L > 0 π₯𝐒𝐦 𝒙→𝒄 𝒇 𝒙 π’ˆ 𝒙 =βˆ’βˆž if L < 0 Quotient: π₯𝐒𝐦 𝒙→𝒄 π’ˆ 𝒙 𝒇 𝒙 =𝟎 C. Similar properties hold for one–sided limits and for functions for which the limit of 𝒇 𝒙 as 𝒙 approaches 𝒄 is +∞. 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

15 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
If π₯𝐒𝐦 π’™β†’βˆ’βˆž 𝒇 𝒙 =𝑳= π₯𝐒𝐦 π’™β†’βˆž 𝒇 𝒙 , then the line π’š = 𝑳 is a horizontal asymptote for the graph of 𝒇(𝒙). There is not a negative left and right bound side. With rational functions and establishing vertical asymptotes, determine the highest exponents on top, bottom, and the same. If 𝒓 is a positive rational number, 𝒄 is any number , and 𝒙𝒓 is defined when 𝒙 < 𝟎 of π₯𝐒𝐦 π’™β†’βˆ’βˆž 𝒄 𝒙 𝒓 =𝟎= π₯𝐒𝐦 π’™β†’βˆž 𝒄 𝒙 𝒓 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

16 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 3 If π₯𝐒𝐦 π’™β†’πŸŽ 𝟏 𝒙 𝟐 =∞ and π₯𝐒𝐦 π’™β†’πŸŽ 𝟏=𝟏, solve π₯𝐒𝐦 π’™β†’πŸŽ 𝟏+ 𝟏 𝒙 𝟐 . 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

17 Example 4 If π₯𝐒𝐦 𝒙→ 𝟏 βˆ’ 𝒙 𝟐 +𝟏 =𝟐 and π₯𝐒𝐦 𝒙→ 𝟏 βˆ’ 𝟏 π’™βˆ’πŸ =βˆ’βˆž, solve π₯𝐒𝐦 𝒙→ 𝟏 βˆ’ 𝒙 𝟐 +𝟏 𝟏/(π’™βˆ’πŸ) . π’ˆ 𝒙 goes first over 𝒇 𝒙 because limit of π’ˆ 𝒙 = constant 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

18 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Your Turn If π₯𝐒𝐦 𝒙→ 𝟏 βˆ’ 𝒙 𝟐 +𝟏 =𝟐 and π₯𝐒𝐦 𝒙→ 𝟏 βˆ’ 𝐜𝐨𝐭 𝝅𝒙 =βˆ’βˆž, solve π₯𝐒𝐦 𝒙→ 𝟏 βˆ’ 𝒙 𝟐 +𝟏 𝐜𝐨𝐭 𝝅𝒙 . 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

19 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 5 Solve π₯𝐒𝐦 𝒙→ πŸ‘ + 𝟐 π’™βˆ’πŸ‘ . 𝒙 𝟐.πŸ— 𝟐.πŸ—πŸ— 𝟐.πŸ—πŸ—πŸ— πŸ‘ πŸ‘.𝟎𝟎𝟏 πŸ‘.𝟎𝟏 πŸ‘.𝟏 𝒇(𝒙) βˆ’πŸπŸŽ βˆ’πŸπŸŽπŸŽ 𝟐𝟎𝟎 𝟐𝟎 βˆ’πŸπŸŽ 𝟐𝟎 βˆ’πŸπŸŽ βˆ’πŸπŸŽπŸŽ βˆ’πŸπŸŽπŸŽπŸŽ ??? 𝟐𝟎𝟎𝟎 𝟐𝟎𝟎 𝟐𝟎 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

20 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 5 Solve π₯𝐒𝐦 𝒙→ πŸ‘ + 𝟐 π’™βˆ’πŸ‘ . 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

21 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Your Turn Solve π₯𝐒𝐦 𝒙→ 𝟏 + 𝒙 𝟐 βˆ’πŸ‘π’™ π’™βˆ’πŸ . 𝒙 𝟏 𝟏.𝟎𝟎𝟏 𝟏.𝟎𝟏 𝟏.𝟏 𝒇(𝒙) βˆ’πŸπŸŽπŸ βˆ’πŸπŸŽ.πŸ— βˆ’πŸπŸŽ.πŸ— ??? βˆ’πŸπŸŽπŸŽπŸ βˆ’πŸπŸŽπŸ βˆ’πŸπŸŽ.πŸ— 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

22 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
When a limit does not exist, it means it approaches towards negative infinity or positive infinity. If a limit does not exist, write β€œDNE” with a reason However, if the limit does not exist and you are asked why, you can respond β€œΒ±βˆžβ€ if it approaches there 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

23 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Steps Establish the highest exponent Divide each term by the biggest exponent of the denominator Same Exponents: Divide the leading coefficients Smaller Exponent/Bigger Exponent: (S/B) = 𝟎 Bigger Exponent/Smaller Exponent: (B/S) = ±∞ Simplify 12/2/ :27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

24 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 6 Solve π₯𝐒𝐦 π’™β†’βˆ’βˆž πŸ‘π’™ 𝟐 𝒙 𝟐 +𝟏 and π₯𝐒𝐦 π’™β†’βˆž πŸ‘π’™ 𝟐 𝒙 𝟐 +𝟏 𝒙 βˆ’βˆž βˆ’πŸπŸŽπŸŽ βˆ’πŸπŸŽ βˆ’πŸ 𝟎 𝟏 𝟏𝟎 𝟏𝟎𝟎 ∞ 𝒇(𝒙) 𝒇(𝒙) 𝟐.πŸ—πŸ—πŸ— 𝟐.πŸ—πŸ• 𝟏.πŸ“ 𝟎 𝒇 𝒙 πŸ‘ 𝟐.πŸ—πŸ—πŸ— 𝟐.πŸ—πŸ• 𝟏.πŸ“ 𝟎 𝒇(𝒙) 𝟐.πŸ—πŸ• 𝟏.πŸ“ 𝟎 𝒇 𝒙 𝟏.πŸ“ 𝟎 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

25 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 6 (Easier Way) Solve π₯𝐒𝐦 π’™β†’βˆ’βˆž πŸ‘π’™ 𝟐 𝒙 𝟐 +𝟏 Same β€œCo” 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

26 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 7 Solve π₯𝐒𝐦 π’™β†’βˆž πŸ” πŸ‘ 𝒙 +πŸ‘ πŸ” 𝒙 +πŸ’ πŸ‘ πŸ–π’™ βˆ’ πŸ” 𝒙 βˆ’πŸπŸŽ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

27 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Your Turn Solve π₯𝐒𝐦 π’™β†’βˆž 𝒙 𝟐 βˆ’πŸ πŸβˆ’πŸπ’™ 𝟐 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

28 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 8 Solve π₯𝐒𝐦 π’™β†’βˆž πŸπ’™βˆ’πŸ 𝒙+𝟏 Same β€œCo” 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

29 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 9 Solve π₯𝐒𝐦 π’™β†’βˆž πŸ‘π’™+πŸ“ πŸ’π’™ 𝟐 +𝟏 𝑺 𝑩 =𝟎 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

30 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 10 Solve π₯𝐒𝐦 π’™β†’βˆ’βˆž πŸ“ 𝒙 πŸ’ βˆ’ πŸπ’™ 𝟐 πŸ’π’™ 𝟐 +𝟏 𝑩 𝑺 =±∞ 𝒙 βˆ’πŸπŸŽπŸŽπŸŽ βˆ’πŸπŸŽπŸŽ β€“πŸ“πŸŽ π’š πŸπŸπŸ’πŸ—πŸ—πŸ—πŸ—.πŸ“ πŸπŸπŸ“πŸŽπŸŽ πŸ‘πŸŽπŸπŸ’.πŸ“ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

31 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Your Turn Solve π₯𝐒𝐦 π’™β†’βˆž 𝟐 𝒙 𝟐 βˆ’πŸ‘ πŸ•π’™+πŸ’ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

32 In Certain Order Identify the following of functions in order of β€œbigness” and how they grow 𝒆𝒙 , π₯𝐧⁑𝒙, 𝒄, π’™πŸ, 𝐬𝐒𝐧⁑𝒙 or πœπ¨π’”β‘π’™ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

33 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 11 Solve π₯𝐒𝐦 π’™β†’βˆž 𝒆 𝒙 π₯𝐧 𝒙 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

34 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 12 Solve π₯𝐒𝐦 π’™β†’βˆž πŸ‘ 𝒙 𝟐 +πŸ” πŸ“βˆ’πŸπ’™ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

35 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 13 Solve π₯𝐒𝐦 π’™β†’βˆ’βˆž 𝟐 𝒙 𝟐 βˆ’πŸ βˆ’πŸ“π’™+πŸ‘ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

36 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Your Turn Solve π₯𝐒𝐦 π’™β†’βˆ’βˆž πŸ‘ 𝒙 𝟐 +πŸ‘ 𝒙+πŸ‘ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

37 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 14 Identify the horizontal asymptote of π’ˆ 𝒙 = πŸ’π’™ 𝟐 +πŸ“ πŸ‘βˆ’πŸπ’™ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

38 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 14 Identify the horizontal asymptote of π’ˆ 𝒙 = πŸ’π’™ 𝟐 +πŸ“ πŸ‘βˆ’πŸπ’™ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

39 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 15 Identify the horizontal asymptote of π’ˆ 𝒙 = 𝒆 πŸπ’™ +𝟏 𝟏+ 𝒆 𝒙 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

40 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 15 Identify the horizontal asymptote of π’ˆ 𝒙 = 𝒆 πŸπ’™ +𝟏 𝟏+ 𝒆 𝒙 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

41 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Your Turn Identify the horizontal asymptote of 𝒉 𝒙 = πŸ—π’™+𝟏 πŸ‘βˆ’πŸ‘π’™ 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

42 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Example 16 Identify the horizontal asymptotes of 𝒇 𝒙 = πŸ’+πŸ” 𝒆 𝒙 πŸ“βˆ’πŸ— 𝒆 𝒙 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

43 AP Multiple Choice Practice Question 1 (non-calculator)
Solve π₯𝐒𝐦 π’•β†’βˆž 𝒕 πŸ’/πŸ‘ + 𝒕 𝟏/πŸ‘ πŸ’ 𝒕 𝟐/πŸ‘ +𝟏 𝟐 (A) βˆ’ 𝟏 πŸπŸ” (B) 𝟏 πŸ’ (C) 𝟎 (D) 𝟏 πŸπŸ” 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

44 AP Multiple Choice Practice Question 1 (non-calculator)
Solve π₯𝐒𝐦 π’•β†’βˆž 𝒕 πŸ’/πŸ‘ + 𝒕 𝟏/πŸ‘ πŸ’ 𝒕 𝟐/πŸ‘ +𝟏 𝟐 Vocabulary Connections and Process Answer and Justifications 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

45 Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity
Assignment Page odd, odd, odd Page all, odd 12/2/ :27 PM Β§1.5 & Β§3.5: Infinite Limits and Limits at Infinity

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