Special Limits - Homepages.math.uic.edu
Special Limitsdefinition of eThe number e is defined as a limit. Here is one definition:1e lim (1 x) xx 0
Special Limitsdefinition of eThe number e is defined as a limit. Here is one definition:1e lim (1 x) xx 0 A good way to evaluate this limit is make a table of numbers.
Special Limitsdefinition of eThe number e is defined as a limit. Here is one definition:1e lim (1 x) xx 0 A good way to evaluate this limit is make a table of numbers.x1(1 x) x.1.010.0010.00010.00001 0
Special Limitsdefinition of eThe number e is defined as a limit. Here is one definition:1e lim (1 x) xx 0 A good way to evaluate this limit is make a table of numbers.x1(1 x) x.12.5937.010.0010.00010.00001 0
Special Limitsdefinition of eThe number e is defined as a limit. Here is one definition:1e lim (1 x) xx 0 A good way to evaluate this limit is make a table of numbers.x1(1 x) x.12.5937.012.704810.0010.00010.00001 0
Special Limitsdefinition of eThe number e is defined as a limit. Here is one definition:1e lim (1 x) xx 0 A good way to evaluate this limit is make a table of numbers.x1(1 x) x.12.5937.012.704810.0012.716920.00010.00001 0
Special Limitsdefinition of eThe number e is defined as a limit. Here is one definition:1e lim (1 x) xx 0 A good way to evaluate this limit is make a table of numbers.x1(1 x) 1 0
Special Limitsdefinition of eThe number e is defined as a limit. Here is one definition:1e lim (1 x) xx 0 A good way to evaluate this limit is make a table of numbers.x1(1 x) 12.71826 0
Special Limitsdefinition of eThe number e is defined as a limit. Here is one definition:1e lim (1 x) xx 0 A good way to evaluate this limit is make a table of numbers.x1(1 x) x.12.5937.012.70481Where e 2.7 1828 1828 · · ·0.0012.716920.00012.718140.000012.71826 0 e
Special Limitsdefinition of eThe number e is defined as a limit. Here is one definition:1e lim (1 x) xx 0 A good way to evaluate this limit is make a table of numbers.x1(1 x) x.12.5937.012.704810.0012.716920.00012.71814Where e 2.7 1828 1828 · · ·This limit will give the same result: e limx 1 1x x0.000012.71826 0 e
Special Limitsalternate definition of eHere is an equivalient definition for e: 1 xe lim 1 x x
Special Limitsalternate definition of eHere is an equivalient definition for e: 1 xe lim 1 x xx1(1 x) x1001000100001000000
Special Limitsalternate definition of eHere is an equivalient definition for e: 1 xe lim 1 x xx1(1 x) x1002.704811000100001000000
Special Limitsalternate definition of eHere is an equivalient definition for e: 1 xe lim 1 x xx1(1 x) x1002.7048110002.71692100001000000
Special Limitsalternate definition of eHere is an equivalient definition for e: 1 xe lim 1 x xx1(1 x) x1002.7048110002.71692100002.718151000000
Special Limitsalternate definition of eHere is an equivalient definition for e: 1 xe lim 1 x xx1(1 x) x1002.7048110002.71692100002.7181510000002.71827
Special Limitsalternate definition of eHere is an equivalient definition for e: 1 xe lim 1 x xx1(1 x) x1002.7048110002.71692100002.7181510000002.71827 e
Special Limitsalternate definition of eHere is an equivalient definition for e: 1 xe lim 1 x xx1(1 x) x1002.7048110002.71692100002.71815Note thatlim f(x)x 0is the same as10000002.71827 e
Special Limitsalternate definition of eHere is an equivalient definition for e: 1 xe lim 1 x xx1(1 x) x1002.7048110002.71692100002.71815Note thatlim f(x)x 0is the same aslim fx 1x10000002.71827 e
Special Limitse the natural baseIthe number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10
Special Limitse the natural baseIthe number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10Ie 2.71828182845904509080 · · ·
Special Limitse the natural baseIthe number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10Ie 2.71828182845904509080 · · ·Ie is a number between 2 and 3. A little closer to 3.
Special Limitse the natural baseIthe number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10Ie 2.71828182845904509080 · · ·Ie is a number between 2 and 3. A little closer to 3.Ie is easy to remember to 9 decimal places because 1828repeats twice: e 2.718281828. For this reason, do not use2.7 to extimate e.
Special Limitse the natural baseIthe number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10Ie 2.71828182845904509080 · · ·Ie is a number between 2 and 3. A little closer to 3.Ie is easy to remember to 9 decimal places because 1828repeats twice: e 2.718281828. For this reason, do not use2.7 to extimate e.Iuse the ex button on your calculator to find e. Use 1 for x.
Special Limitse the natural baseIthe number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10Ie 2.71828182845904509080 · · ·Ie is a number between 2 and 3. A little closer to 3.Ie is easy to remember to 9 decimal places because 1828repeats twice: e 2.718281828. For this reason, do not use2.7 to extimate e.Iuse the ex button on your calculator to find e. Use 1 for x.Iexample: F Pert is often used for calculating compoundinterest in business applications.
Special Limitsf(x) 1xHere are four useful limits:Ilimx 1x 0 0
Special Limitsf(x) 1xHere are four useful limits:Ilimx Ilimx 1x1x 0 0 0 0
Special Limitsf(x) 1xHere are four useful limits:Ilimx Ilimx Ilimx 0 1x1x1x 0 0 0 0
Special Limitsf(x) 1xHere are four useful limits:Ilim1xx Ilim1x Ilimx 0 Ilimx 0 x1x1x 0 0 0 0
Special Limitsf(x) 1xcont.The general idea is this:I1 BIG small
Special Limitsf(x) 1xcont.The general idea is this:I1 BIGI1 BIG small small
Special Limitsf(x) 1xcont.The general idea is this:I1 BIGI1 BIGI small small1 small BIG
Special Limitsf(x) 1xcont.The general idea is this:I1 BIGI1 BIGI small small1 smallI1 small BIG BIG
Special Limitsf(x) 1xf 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 8.5 7.0 5.5 4.0 2.5 1.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1.0 2.5 4.0 5.5 7.0 8.5x
Special Limits1BIG small,1small BIGLook at 1x using some numbers:x10 100 1000 10000f(x) 1x.1 .01 .001 .0001xf(x) 1x-10-.1xf(x) 1x.110xf(x) .00001.00001100000-.0001-10000 0 0 0 -.00001-100000 0
Special Limitsx for PolynomialsIwhen taking limits of polynomials to drop the lowerdegree terms and only keep the higest degree term of of thepolymomial.
Special Limitsx for PolynomialsIwhen taking limits of polynomials to drop the lowerdegree terms and only keep the higest degree term of of thepolymomial.Ithis is an intermediate step in taking the limit. Use algebra tosimplify the expression at this step then continue to work onfinding the limit to infinity.
Special Limitsx for PolynomialsIwhen taking limits of polynomials to drop the lowerdegree terms and only keep the higest degree term of of thepolymomial.Ithis is an intermediate step in taking the limit. Use algebra tosimplify the expression at this step then continue to work onfinding the limit to infinity.Ithis works because for large values of x the highest powerterm of the polynomial is so much larger that all of thesmaller degree terms that the smaller degree terms have noeffect in the limit to infinity.
Special Limitsexamples of x for PolynomialsIlimx 2x3 99x 10001 2x2 4x3
Special Limitsexamples of x for PolynomialsIlimx 2x3 99x 10001 2x2 4x3
Special Limitsexamples of x for PolynomialsIlimx 2x3 99x 10001 2x2 4x3 limx 2x34x3
Special Limitsexamples of x for PolynomialsIlimx 2x3 99x 10001 2x2 4x3 limx 2x34x3 limx 24
Special Limitsexamples of x for PolynomialsI2x3 99x 10001 2x2 4x31limx 2limx limx 2x34x3 limx 24
Special Limitsexamples of x for PolynomialsI2x3 99x 10001 2x2 4x31limx 2 12limx limx 2x34x3 limx 24
Special Limitsexample of x for Polynomialslimx 2x3 99x 10001 2x2 4x3
Special Limitsexample of x for Polynomialslimx 2x3 99x 10001 2x2 4x3 limx 2x34x3
Special Limitsexample of x for Polynomialslimx 2x3 99x 10001 2x2 4x3 limx 2x34x3 limx 24
Special Limitsexample of x for Polynomialslimx 2x3 99x 10001 2x2 4x3 limx 2x34x3 limx 24 limx 12
Special Limitsexample of x for Polynomialslimx 2x3 99x 10001 2x2 4x3 limx 2x34x3 limx 24 limx 12 12
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x3
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x3 limx 2x44x3
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x3 limx 2x44x3 limx 42 x
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x3 limx 2x44x3 limx 42 x limx 21 x
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x3 limx 2x44x3 limx 42 x limx 21 x 12limx x
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x3 limx 2x44x3 limx 42 x limx 21 x 12limx x 12·
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x3 limx 2x44x3 limx 42 x limx 21 x 12limx x 12·
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6 limx 24x2
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6 limx 24x2 limx 12x2
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6 limx 24x2 limx 12x2 12limx 1x2
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6 limx 24x2 limx 12x2 12limx 12·1 1x2
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6 limx 24x2 limx 12x2 12limx 1x2 12· 12· 0 0 01
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6 limx 24x2 limx 12x2 12limx 1x2 12· 12· 0 0 01
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6 limx 24x2 limx 12x2 12limx 1x2 12· 12· 0 0 01
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6 limx 24x2 limx 12x2 12limx 1x2 12· 12· 0 0 01
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6 limx 24x2 limx 12x2 12limx 1x2 12· 12· 0 0 01
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6 limx 24x2 limx 12x2 12limx 1x2 12· 12· 0 0 01
Special Limitsexample of x for Polynomialslimx 2x4 99x 10001 2x2 4x6 limx 2x44x6 limx 24x2 limx 12x2 12limx 1x2 12· 12· 0 0 01
Special Limits e the natural base I the number e is the natural base in calculus. Many expressions in calculus are simpler in base e than in other bases like base 2 or base 10 I e 2:71828182845904509080 I e is a number between 2 and 3. A little closer to 3. I e is easy to remember to 9 de
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