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# Note for MATH 263A with Professor Holston at Ohio-section 1.6 notes

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Date Created: 02/06/15

Limits invoivmg infinity Chris Hoiston Limits Involving Infinity Chris Holston lelts lnvolvmg lnfinm Vertical Asymptotes cm Holston Preliminaries We write lim fx 00 when as X approaches the XHa number a from the left the function becomes increasingly large lelts lnvolvmg lnfinm Vertical Asymptotes Chris Holston WWW We write lim fx 00 when as X approaches the Xaa number a from the left the function becomes increasingly large lim fx 700 means almost the same thing The XHE function becomes increasingly large in magnitude except the function values are all negative lelts lnvolvmg lnfinm Vertical Asymptotes Chris Holston WWW We write lim fx 00 when as X approaches the Xaa number a from the left the function becomes increasingly large lim fx 700 means almost the same thing The XHE function becomes increasingly large in magnitude except the function values are all negative limJr fx ioo are Xgta defined similarly lelts lnvolvmg lnfinm Vertical Asymptotes Chris Holston WWW We write lim fx 00 when as X approaches the Xaa number a from the left the function becomes increasingly large lim fx 700 means almost the same thing The XHE function becomes increasingly large in magnitude except the function values are all negative limJr fx ioo are Xgta defined similarly Definition We say the line X a is a vertical asymptote for fx if either lim fx ioo or lingr fx ioo XH XHa lelts lnvolvmg lnfinm Vertical Asymptotes Chris Holston WWW We write lim fx 00 when as X approaches the Xaa number a from the left the function becomes increasingly large lim fx 700 means almost the same thing The XHE function becomes increasingly large in magnitude except the function values are all negative limJr fx ioo are Xgta defined similarly Definition We say the line X a is a vertical asymptote for fx if either lim fx ioo or lingr fx ioo XH XHa Remarks Nothing in this definition requires that a is not in the domain of fx lelts lnvolvmg lnfinm Vertical Asymptotes Chris Holston WWW We write lim fx 00 when as X approaches the Xaa number a from the left the function becomes increasingly large lim fx 700 means almost the same thing The XHE function becomes increasingly large in magnitude except the function values are all negative limJr fx ioo are Xgta defined similarly Definition We say the line X a is a vertical asymptote for fx if either lim fx ioo or lingr fx ioo XH XHa Remarks Nothing in this definition requires that a is not in the domain of fx The function need not approach the vertical asymptote from both sides lelts lnvolvmg lnfinm Vertical Asymptotes Chris Holston WWW We write lim fx 00 when as X approaches the Xaa number a from the left the function becomes increasingly large lim fx 700 means almost the same thing The XHE function becomes increasingly large in magnitude except the function values are all negative limJr fx ioo are Xgta defined similarly Definition We say the line X a is a vertical asymptote for fx if either lim fx ioo or lingr fx ioo XH XHa Remarks Nothing in this definition requires that a is not in the domain of fx The function need not approach the vertical asymptote from both sides A function can have infinitely many vertical asymptotes EX fx tanX Limits invoivmg infinity Two sided Limits Chris Hoiston Preiiminzries Iim fx ioo are defined in the same way as finite limits xgta Limits invoivmg infinity Two sided Limits Chris Hoiston Preiiminzries Iim fx ioo are defined in the same way as finite limits xgta 1 Examples IIm 7 oo xaox Limits invoivmg infinity Two sided Limits Chris Hoiston Preiiminzries Iim fx ioo are defined in the same way as finite limits xgta 1 Examples IIm 7 oo xaox l DNE Iim xaox umus nvohmg n nm Horizontal Asymptotes cm Ho ston Prehmmznes im fx L means as we take increasingly large values Xgtoo of X to the quotfar right of the number line the function approaches the real number L lelts lnvolvmg lnfinm Cm Hm Horizontal Asymptotes Preliminaries lim fx L means as we take increasingly large values of X to the quotfar right of the number line the function approaches the real number L lim fx L is defined similarly lelts lnvolvmg lnfinm Horizontal Asymptotes Chris Holston Preliminaries lim fx L means as we take increasingly large values Xgtoo of X to the quotfar right of the number line the function approaches the real number L lim fx L is defined Xaioo similarly Definition We say the line y L is a horizontal asymptote for fx is either lim fx L or lim fx L Xgtoo lelts lnvolvmg lnfinm Horizontal Asymptotes cm Holston Preliminaries lim fx L means as we take increasingly large values of X to the quotfar right of the number line the function approaches the real number L lim fx L is defined similarly Definition We say the line y L is a horizontal asymptote for fx is either lim fx L or lim fx L Xgtoo Remarks Nothing in this definition requires that the function can never attain the value L lelts lnvolvmg lnfinm Horizontal Asymptotes cm Holston Preliminaries lim fx L means as we take increasingly large values Xgtoo of X to the quotfar right of the number line the function approaches the real number L lim fx L is defined similarly Definition We say the line y L is a horizontal asymptote for fx is either lim fx L or lim fx L Xgtoo Remarks Nothing in this definition requires that the function can never attain the value L The function need not approach the horizontal asymptote from both sides lelts lnvolvmg lnfinm Horizontal Asymptotes cm Holston Preliminaries lim fx L means as we take increasingly large values of X to the quotfar right of the number line the function approaches the real number L lim fx L is defined similarly Definition We say the line y L is a horizontal asymptote for fx is either lim fx L or lim fx L Xgtoo Remarks Nothing in this definition requires that the function can never attain the value L The function need not approach the horizontal asymptote from both sides A function can have no more than 2 horizontal asymptotes Limits invoivmg infinity Limits at Infinity Chris Hoiston Preiiminzries Not all functions have horizontal asymptotes Examples Iim x2 00 Xgtoo Limits invoivmg infinity Limits at Infinity Chris Hoiston Preiiminzries Not all functions have horizontal asymptotes Examples Iim x2 00 Xgtoo Iim x2 oo Xaioo Limits invoivmg infinity Limits at Infinity Chris Hoiston Preiiminzries Not all functions have horizontal asymptotes Examples Iim x2 00 Xgtoo Iim x2 oo Xaioo Iim 7X3 foo Xgtoo Limits invoivmg infinity Limits at Infinity Chris Hoiston Preiiminzries Not all functions have horizontal asymptotes Examples Iim x2 00 Xgtoo Iim x2 oo Xaioo Iim 7X3 foo Xgtoo Iim 7X3 oo xaioo Limits invoivmg infinity Limits at Infinity Chris Hoiston Preiiminzries Not all functions have horizontal asymptotes Examples Iim x2 00 Xgtoo Iim x2 oo Xaioo Iim 7X3 foo Xgtoo Iim 7X3 oo Xaioo Iim sinx DNE Xgtoo errts lmolvmg ln nm Calculating LImIts lnvolvmg lnflnlty cm Holston Prellmmzrles The following properties illustrates that O and 00 are like inverses Limits lnvolvrng infinity Calculating Limits Involving Infinity Chris Holston Pieiiiiiiiiziies The following properties illustrates that O and 00 are like inverses 1 If 11 fXioo then Iim O Xaioo m Limits lnvolvrng infinity WWW Calculating Limits Involving Infinity Pieiiiiiiiiziies The following properties illustrates that O and 00 are like inverses 1 If 11 fxioo then Iim O Xaioo m If Iim fx 0 then Iim L ioo and Xgta XHa f X IimJr fx ioo although Iim fx might still not exist Limits invoivmg infinity Chris Hoiston 8 Exerci ses Sketch the graph of an example of a function 2 that satisfies all of the given conditions Iim fx foo Iim fx 2f0 07 f is even Xgt3 14 Fi d the limit Jim 55 x w 5 Aghl b 14 Fi d the limit Hm 55 x w 5 4amp9 Aghl b1 b 16 4gu452 E 16 4gu452 E 18 Fii d he limit 5 Aghl b1 b 18 4gu452 E 4gu452 E Limits invoivmg infinity Chris Hoiston Exerci 595 Find the limit Ii t2 2 m taioot3t271 0 Limits invoivmg infinity Chris Hoiston Exerci 595 Find the limit Iim X2 3X7 X2 bx Limits invoivmg infinity Chris Hoiston Exerci 595 Find the limit Iim X2 3X7 X2 bx aib 2

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