Evaluating Limits Analytically Section 12.2 Precalculus PreAP/Dual, Revised ©2017 viet.dang@humbleisd.net 4/20/2019 8:57 AM 12.2: Evaluating Limits

“0/0” Limits AKA: Indeterminate Form Always begin with direct substitution Completely factor the problem Simplify and/or Cancel by identifying a function 𝒈 that agrees with for all x except 𝒙 = 𝒄. Take the limit of 𝒈 Apply algebra rules If necessary, Rationalize the numerator Plug in 𝒙 of the function to get the limit 4/20/2019 8:57 AM 12.2: Evaluating Limits

Example 1 Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒 What form is this? 12.2: Evaluating Limits

Example 1 Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒 AS X APPROACHES 4, f(x) OR Y APPROACHES 8. 4/20/2019 8:57 AM 12.2: Evaluating Limits

Example 1 (Calculator) Solve 𝐥𝐢𝐦 𝒙→𝟒 𝒙 𝟐 −𝟏𝟔 𝒙−𝟒 4/20/2019 8:57 AM 12.2: Evaluating Limits

Your Turn Solve 𝐥𝐢𝐦 𝒙→−𝟑 𝒙 𝟐 +𝒙−𝟔 𝒙+𝟑 4/20/2019 8:57 AM 12.2: Evaluating Limits

Example 2 Solve 𝐥𝐢𝐦 𝒙→−𝟏 𝟐𝒙 𝟐 −𝒙−𝟑 𝒙 𝟐 −𝟐𝒙−𝟑 4/20/2019 8:57 AM 12.2: Evaluating Limits

Example 3 Solve 𝐥𝐢𝐦 𝒙→ 𝝅 𝟐 𝐭𝐚𝐧 𝒙 𝐜𝐨𝐬 𝒙 4/20/2019 8:57 AM 12.2: Evaluating Limits

Your Turn Solve 𝐥𝐢𝐦 𝒕→−𝟏 𝒕 𝟑 −𝒕 𝒕 𝟐 −𝟏 4/20/2019 8:57 AM 12.2: Evaluating Limits

When in Algebra… NO RADICALS IN THE DENOMINATOR You learned to: NO RADICALS IN THE DENOMINATOR FOR LIMITS, NO RADICALS IN THE NUMERATOR 4/20/2019 8:57 AM 12.2: Evaluating Limits

Example 4 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 What form is this? 4/20/2019 8:57 AM 12.2: Evaluating Limits

Example 4 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 NO NEED TO FOIL THE BOTTOM 12.2: Evaluating Limits

Example 4 Solve 𝐥𝐢𝐦 𝒙→𝟗 𝒙 −𝟑 𝒙−𝟗 4/20/2019 8:57 AM 12.2: Evaluating Limits

Example 5 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝒙+𝟏 −𝟏 𝒙 4/20/2019 8:57 AM 12.2: Evaluating Limits

Your Turn Solve 𝐥𝐢𝐦 𝒙→−𝟑 𝒙+𝟕 −𝟐 𝒙+𝟑 . Hint: Don’t combine like terms to the denominator, too early 4/20/2019 8:57 AM 12.2: Evaluating Limits

Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 What form is this? 12.2: Evaluating Limits

Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 4/20/2019 8:57 AM 12.2: Evaluating Limits

Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 4/20/2019 8:57 AM 12.2: Evaluating Limits

Example 6 Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝟓+𝒙 − 𝟏 𝟓 𝒙 4/20/2019 8:57 AM 12.2: Evaluating Limits

Your Turn Solve 𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝒙+𝟒 − 𝟏 𝟒 𝒙 4/20/2019 8:57 AM 12.2: Evaluating Limits

Assignment Page 833 5-7, 9-21 odd 4/20/2019 8:57 AM 12.2: Evaluating Limits