L2 Algebraic Limits - Part 3
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Unit 1B
Limits
Lesson 2: Algebraic Limits
The limits of algebraic functions involving radicals can also be determined using algebraic analysis. A useful technique for evaluating such limits is to multiply by the conjugate of the numerator or denominator to rationalize part of the function.
For example, the conjugate of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msqrt»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«/msqrt»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msqrt»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math».
Evaluate «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»t«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«mi»t«/mi»«/mfrac»«/mrow»«/mstyle»«/math».
| In general, the conjugate of a binomial is the same expression with a different sign between the terms. For example, the conjugate of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»-«/mo»«mn»5«/mn»«/mstyle»«/math». |
The substitution of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»t«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» results in «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»t«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«msqrt»«msup»«mn»0«/mn»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«mn»0«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mn»0«/mn»«mn»0«/mn»«/mfrac»«/mrow»«/mstyle»«/math».
As this is of indeterminate form, a method other than substitution must be used. Since the numerator is not factorable, multiply both the numerator and denominator by the conjugate of the radical expression in the numerator, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math», to rationalize the numerator.
The limit is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math».
As this is of indeterminate form, a method other than substitution must be used. Since the numerator is not factorable, multiply both the numerator and denominator by the conjugate of the radical expression in the numerator, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math», to rationalize the numerator.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»t«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mi
mathvariant=¨normal¨»(«/mi»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mi»t«/mi»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨
mathcolor=¨#0080FF¨»(«/mi»«msqrt mathcolor=¨#0080FF¨»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo mathcolor=¨#0080FF¨»+«/mo»«mn mathcolor=¨#0080FF¨»2«/mn»«mi mathvariant=¨normal¨ mathcolor=¨#0080FF¨»)«/mi»«/mrow»«mrow»«mi
mathvariant=¨normal¨ mathcolor=¨#0080FF¨»(«/mi»«msqrt mathcolor=¨#0080FF¨»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo mathcolor=¨#0080FF¨»+«/mo»«mn mathcolor=¨#0080FF¨»2«/mn»«mi mathvariant=¨normal¨ mathcolor=¨#0080FF¨»)«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»t«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»+«/mo»«mn»2«/mn»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»2«/mn»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»4«/mn»«/mrow»«mrow»«mi»t«/mi»«mfenced»«mrow»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»t«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mo»§#8722;«/mo»«mn»4«/mn»«/mrow»«mrow»«mi»t«/mi»«mfenced»«mrow»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»t«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mrow»«mi»t«/mi»«mi
mathvariant=¨normal¨»(«/mi»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»+«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»t«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«msup»«mi»t«/mi»«menclose
notation=¨updiagonalstrike¨»«mn»2«/mn»«/menclose»«/msup»«mrow»«menclose notation=¨updiagonalstrike¨»«mi»t«/mi»«/menclose»«mi mathvariant=¨normal¨»(«/mi»«msqrt»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»+«/mo»«mn»2«/mn»«mi
mathvariant=¨normal¨»)«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»0«/mn»«mrow»«msqrt»«msup»«mn»0«/mn»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«/msqrt»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»0«/mn»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
| Use the distributive property in the numerator, simplify, and then collect like terms. |
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