Infinite Limits and Limits at Infinity Section 1.5 and 3.5 Calculus BC AP/Dual, Revised ©2017 viet.dang@humbleisd.net 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §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 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) =±∞, does not exist. The statement, “𝒙 approaches to infinity” really describes the limit to reach a constant number. Let 𝒇 and 𝒈 be continuous on an open interval containing 𝒄. If 𝒇(𝒄)≠ 𝟎, 𝒈(𝒄) = 𝟎, then 𝒉 𝒙 = 𝒇 𝒙 𝒈 𝒙 has a vertical asymptote at 𝒙=𝒄. 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §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. 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

Review Determine 𝐥𝐢𝐦 𝒙→ 𝟐 − 𝟑 𝒙−𝟐 and 𝐥𝐢𝐦 𝒙→ 𝟐 + 𝟑 𝒙−𝟐 𝒙 𝟏.𝟗 𝟏.𝟗𝟗 𝟏.𝟗𝟗𝟗 𝟐 𝟐.𝟎𝟎𝟏 𝟐.𝟎𝟏 𝟐.𝟏 𝒇(𝒙) 𝒇(𝒙) −𝟑𝟎 −𝟑𝟎𝟎 −𝟑𝟎𝟎𝟎 𝟑𝟎𝟎𝟎 𝟑𝟎𝟎 𝟑𝟎 𝒇(𝒙) −𝟑𝟎 𝟑𝟎 𝒇(𝒙) –𝟑𝟎 –𝟑𝟎𝟎 𝟑𝟎𝟎 𝟑𝟎 𝒇 𝒙 DECREASES without bound 𝒇 𝒙 INCREASES without bound 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Review Determine 𝐥𝐢𝐦 𝒙→ 𝟐 − 𝟑 𝒙−𝟐 and 𝐥𝐢𝐦 𝒙→ 𝟐 + 𝟑 𝒙−𝟐 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 1 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟏 𝟐 𝒙−𝟏 and 𝐥𝐢𝐦 𝒙→𝟏 = 𝟏 𝟐 𝒙−𝟏 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 1 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟏 𝟐 𝒙−𝟏 and 𝐥𝐢𝐦 𝒙→𝟏 = 𝟏 𝟐 𝒙−𝟏 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 1 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟏 𝟐 𝒙−𝟏 and 𝐥𝐢𝐦 𝒙→𝟏 = 𝟏 𝟐 𝒙−𝟏 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 2 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟐 𝐬𝐢𝐧 𝒙 and 𝐥𝐢𝐦 𝒙→𝟎 = 𝟐 𝐬𝐢𝐧 𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 2 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟐 𝐬𝐢𝐧 𝒙 and 𝐥𝐢𝐦 𝒙→𝟎 = 𝟐 𝐬𝐢𝐧 𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 2 Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟐 𝐬𝐢𝐧 𝒙 and 𝐥𝐢𝐦 𝒙→𝟎 = 𝟐 𝐬𝐢𝐧 𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Your Turn Solve the vertical asymptotes (if any) of 𝒇 𝒙 = 𝟏 𝒙 𝟐 and 𝐥𝐢𝐦 𝒙→𝟎 = 𝟏 𝒙 𝟐 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §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 𝐥𝐢𝐦 𝒙→𝒄 𝒇(𝒙) =±∞, does not exist. The statement, “𝒙 approaches to infinity” really describes the limit to reach a constant number. Let 𝒇 and 𝒈 be continuous on an open interval containing 𝒄. If 𝒇(𝒄)≠ 𝟎, 𝒈(𝒄) = 𝟎, then 𝒉 𝒙 = 𝒇 𝒙 𝒈 𝒙 has a vertical asymptote at 𝒙=𝒄. 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

Properties of Infinite Limits As we let 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 =∞ and 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 =𝑳 : Types: Sum or difference: 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 ±𝒈 𝒙 =∞±𝑳=∞ Product: 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 =∞ if 𝐥𝐢𝐦 𝒙→𝒄 𝒈 𝒙 or L > 0 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 =−∞ if L < 0 Quotient: 𝐥𝐢𝐦 𝒙→𝒄 𝒈 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 +∞. 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity If 𝐥𝐢𝐦 𝒙→−∞ 𝒇 𝒙 =𝑳= 𝐥𝐢𝐦 𝒙→∞ 𝒇 𝒙 , then the line 𝒚 = 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 𝐥𝐢𝐦 𝒙→−∞ 𝒄 𝒙 𝒓 =𝟎= 𝐥𝐢𝐦 𝒙→∞ 𝒄 𝒙 𝒓 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 3 If 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝒙 𝟐 =∞ and 𝐥𝐢𝐦 𝒙→𝟎 𝟏=𝟏, solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏+ 𝟏 𝒙 𝟐 . 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

Example 4 If 𝐥𝐢𝐦 𝒙→ 𝟏 − 𝒙 𝟐 +𝟏 =𝟐 and 𝐥𝐢𝐦 𝒙→ 𝟏 − 𝟏 𝒙−𝟏 =−∞, solve 𝐥𝐢𝐦 𝒙→ 𝟏 − 𝒙 𝟐 +𝟏 𝟏/(𝒙−𝟏) . 𝒈 𝒙 goes first over 𝒇 𝒙 because limit of 𝒈 𝒙 = constant 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Your Turn If 𝐥𝐢𝐦 𝒙→ 𝟏 − 𝒙 𝟐 +𝟏 =𝟐 and 𝐥𝐢𝐦 𝒙→ 𝟏 − 𝐜𝐨𝐭 𝝅𝒙 =−∞, solve 𝐥𝐢𝐦 𝒙→ 𝟏 − 𝒙 𝟐 +𝟏 𝐜𝐨𝐭 𝝅𝒙 . 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 5 Solve 𝐥𝐢𝐦 𝒙→ 𝟑 + 𝟐 𝒙−𝟑 . 𝒙 𝟐.𝟗 𝟐.𝟗𝟗 𝟐.𝟗𝟗𝟗 𝟑 𝟑.𝟎𝟎𝟏 𝟑.𝟎𝟏 𝟑.𝟏 𝒇(𝒙) −𝟐𝟎 −𝟐𝟎𝟎 𝟐𝟎𝟎 𝟐𝟎 −𝟐𝟎 𝟐𝟎 −𝟐𝟎 −𝟐𝟎𝟎 −𝟐𝟎𝟎𝟎 ??? 𝟐𝟎𝟎𝟎 𝟐𝟎𝟎 𝟐𝟎 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 5 Solve 𝐥𝐢𝐦 𝒙→ 𝟑 + 𝟐 𝒙−𝟑 . 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Your Turn Solve 𝐥𝐢𝐦 𝒙→ 𝟏 + 𝒙 𝟐 −𝟑𝒙 𝒙−𝟏 . 𝒙 𝟏 𝟏.𝟎𝟎𝟏 𝟏.𝟎𝟏 𝟏.𝟏 𝒇(𝒙) −𝟐𝟎𝟏 −𝟐𝟎.𝟗 −𝟐𝟎.𝟗 ??? −𝟐𝟎𝟎𝟏 −𝟐𝟎𝟏 −𝟐𝟎.𝟗 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §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 “DNE” with a reason However, if the limit does not exist and you are asked why, you can respond “±∞” if it approaches there 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §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 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 6 Solve 𝐥𝐢𝐦 𝒙→−∞ 𝟑𝒙 𝟐 𝒙 𝟐 +𝟏 and 𝐥𝐢𝐦 𝒙→∞ 𝟑𝒙 𝟐 𝒙 𝟐 +𝟏 𝒙 −∞ −𝟏𝟎𝟎 −𝟏𝟎 −𝟏 𝟎 𝟏 𝟏𝟎 𝟏𝟎𝟎 ∞ 𝒇(𝒙) 𝒇(𝒙) 𝟐.𝟗𝟗𝟗 𝟐.𝟗𝟕 𝟏.𝟓 𝟎 𝒇 𝒙 𝟑 𝟐.𝟗𝟗𝟗 𝟐.𝟗𝟕 𝟏.𝟓 𝟎 𝒇(𝒙) 𝟐.𝟗𝟕 𝟏.𝟓 𝟎 𝒇 𝒙 𝟏.𝟓 𝟎 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 6 (Easier Way) Solve 𝐥𝐢𝐦 𝒙→−∞ 𝟑𝒙 𝟐 𝒙 𝟐 +𝟏 Same “Co” 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 7 Solve 𝐥𝐢𝐦 𝒙→∞ 𝟔 𝟑 𝒙 +𝟑 𝟔 𝒙 +𝟒 𝟑 𝟖𝒙 − 𝟔 𝒙 −𝟏𝟎 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Your Turn Solve 𝐥𝐢𝐦 𝒙→∞ 𝒙 𝟐 −𝟏 𝟏−𝟐𝒙 𝟐 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 8 Solve 𝐥𝐢𝐦 𝒙→∞ 𝟐𝒙−𝟏 𝒙+𝟏 Same “Co” 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 9 Solve 𝐥𝐢𝐦 𝒙→∞ 𝟑𝒙+𝟓 𝟒𝒙 𝟐 +𝟏 𝑺 𝑩 =𝟎 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 10 Solve 𝐥𝐢𝐦 𝒙→−∞ 𝟓 𝒙 𝟒 − 𝟐𝒙 𝟐 𝟒𝒙 𝟐 +𝟏 𝑩 𝑺 =±∞ 𝒙 −𝟏𝟎𝟎𝟎 −𝟏𝟎𝟎 –𝟓𝟎 𝒚 𝟏𝟐𝟒𝟗𝟗𝟗𝟗.𝟓 𝟏𝟐𝟓𝟎𝟎 𝟑𝟎𝟐𝟒.𝟓 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Your Turn Solve 𝐥𝐢𝐦 𝒙→∞ 𝟐 𝒙 𝟐 −𝟑 𝟕𝒙+𝟒 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

In Certain Order Identify the following of functions in order of “bigness” and how they grow 𝒆𝒙 , 𝐥𝐧⁡𝒙, 𝒄, 𝒙𝟐, 𝐬𝐢𝐧⁡𝒙 or 𝐜𝐨𝒔⁡𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 11 Solve 𝐥𝐢𝐦 𝒙→∞ 𝒆 𝒙 𝐥𝐧 𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 12 Solve 𝐥𝐢𝐦 𝒙→∞ 𝟑 𝒙 𝟐 +𝟔 𝟓−𝟐𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 13 Solve 𝐥𝐢𝐦 𝒙→−∞ 𝟐 𝒙 𝟐 −𝟏 −𝟓𝒙+𝟑 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Your Turn Solve 𝐥𝐢𝐦 𝒙→−∞ 𝟑 𝒙 𝟐 +𝟑 𝒙+𝟑 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 14 Identify the horizontal asymptote of 𝒈 𝒙 = 𝟒𝒙 𝟐 +𝟓 𝟑−𝟐𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 14 Identify the horizontal asymptote of 𝒈 𝒙 = 𝟒𝒙 𝟐 +𝟓 𝟑−𝟐𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 15 Identify the horizontal asymptote of 𝒈 𝒙 = 𝒆 𝟐𝒙 +𝟏 𝟏+ 𝒆 𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 15 Identify the horizontal asymptote of 𝒈 𝒙 = 𝒆 𝟐𝒙 +𝟏 𝟏+ 𝒆 𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Your Turn Identify the horizontal asymptote of 𝒉 𝒙 = 𝟗𝒙+𝟏 𝟑−𝟑𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Example 16 Identify the horizontal asymptotes of 𝒇 𝒙 = 𝟒+𝟔 𝒆 𝒙 𝟓−𝟗 𝒆 𝒙 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

AP Multiple Choice Practice Question 1 (non-calculator) Solve 𝐥𝐢𝐦 𝒕→∞ 𝒕 𝟒/𝟑 + 𝒕 𝟏/𝟑 𝟒 𝒕 𝟐/𝟑 +𝟏 𝟐 (A) − 𝟏 𝟏𝟔 (B) 𝟏 𝟒 (C) 𝟎 (D) 𝟏 𝟏𝟔 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

AP Multiple Choice Practice Question 1 (non-calculator) Solve 𝐥𝐢𝐦 𝒕→∞ 𝒕 𝟒/𝟑 + 𝒕 𝟏/𝟑 𝟒 𝒕 𝟐/𝟑 +𝟏 𝟐 Vocabulary Connections and Process Answer and Justifications 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity

§1.5 & §3.5: Infinite Limits and Limits at Infinity Assignment Page 88 1-7 odd, 13-19 odd, 33-47 odd Page 202 1-6 all, 19-31 odd 12/2/2018 10:27 PM §1.5 & §3.5: Infinite Limits and Limits at Infinity