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Trazado el gráfico de la función/ exp(1/x)/(x^2(exp(1/x)-1)^2)

Gráfico de la función y = exp(1/x)/(x^2(exp(1/x)-1)^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
             1     
             -     
             x     
            e      
f(x) = ------------
                  2
          / 1    \ 
          | -    | 
        2 | x    | 
       x *\e  - 1/
f(x)=e1xx2(e1x−1)2f{\left(x \right)} = \frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} 
f = exp(1/x)/((x^2*(exp(1/x) - 1)^2))
Gráfico de la función
02468-8-6-4-2-101002
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
e1xx2(e1x−1)2=0\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en exp(1/x)/((x^2*(exp(1/x) - 1)^2)).
e1002(−1+e10)2\frac{e^{\frac{1}{0}}}{0^{2} \left(-1 + e^{\frac{1}{0}}\right)^{2}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
−1x2(e1x−1)2e1xx2+(−2x(e1x−1)2+2(e1x−1)e1x)e1xx4(e1x−1)4=0- \frac{\frac{1}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} e^{\frac{1}{x}}}{x^{2}} + \frac{\left(- 2 x \left(e^{\frac{1}{x}} - 1\right)^{2} + 2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{4}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=36512.8341459926x_{1} = 36512.8341459926
x2=13627.9152059125x_{2} = 13627.9152059125
x3=−38924.3992160704x_{3} = -38924.3992160704
x4=−17734.5572561786x_{4} = -17734.5572561786
x5=16170.6287045655x_{5} = 16170.6287045655
x6=23798.8964012984x_{6} = 23798.8964012984
x7=41598.4307745321x_{7} = 41598.4307745321
x8=−40619.5984694662x_{8} = -40619.5984694662
x9=12780.3528127123x_{9} = 12780.3528127123
x10=26341.6742356719x_{10} = 26341.6742356719
x11=30579.6501812154x_{11} = 30579.6501812154
x12=10237.7072058338x_{12} = 10237.7072058338
x13=28884.4582101405x_{13} = 28884.4582101405
x14=39055.6315231319x_{14} = 39055.6315231319
x15=−28753.2265233594x_{15} = -28753.2265233594
x16=28036.8629728761x_{16} = 28036.8629728761
x17=−12649.1267832051x_{17} = -12649.1267832051
x18=−39771.9987445352x_{18} = -39771.9987445352
x19=−29600.822196685x_{19} = -29600.822196685
x20=−41467.1983788078x_{20} = -41467.1983788078
x21=20408.5394735307x_{21} = 20408.5394735307
x22=33970.0390640156x_{22} = 33970.0390640156
x23=−30448.4183467446x_{23} = -30448.4183467446
x24=15323.0538797709x_{24} = 15323.0538797709
x25=18713.3691647864x_{25} = 18713.3691647864
x26=35665.2355060803x_{26} = 35665.2355060803
x27=14475.4824993254x_{27} = 14475.4824993254
x28=33122.4413047548x_{28} = 33122.4413047548
x29=22103.7156359971x_{29} = 22103.7156359971
x30=−32143.6119262135x_{30} = -32143.6119262135
x31=−23667.6653632422x_{31} = -23667.6653632422
x32=25494.0808512544x_{32} = 25494.0808512544
x33=−27905.6313702123x_{33} = -27905.6313702123
x34=−38076.7998971915x_{34} = -38076.7998971915
x35=−8411.44815555337x_{35} = -8411.44815555337
x36=−14344.2549142001x_{36} = -14344.2549142001
x37=−31296.0149348033x_{37} = -31296.0149348033
x38=−18582.1393762791x_{38} = -18582.1393762791
x39=37360.4330398325x_{39} = 37360.4330398325
x40=7695.1720429073x_{40} = 7695.1720429073
x41=32274.8438856766x_{41} = 32274.8438856766
x42=−5869.07851287475x_{42} = -5869.07851287475
x43=−10106.4851277259x_{43} = -10106.4851277259
x44=8542.66540404055x_{44} = 8542.66540404055
x45=−35534.0033480628x_{45} = -35534.0033480628
x46=34817.6371386206x_{46} = 34817.6371386206
x47=6000.27930908944x_{47} = 6000.27930908944
x48=11085.2472232554x_{48} = 11085.2472232554
x49=−15191.8257038658x_{49} = -15191.8257038658
x50=42446.0308832849x_{50} = 42446.0308832849
x51=−34686.4050248436x_{51} = -34686.4050248436
x52=−26210.4428263164x_{52} = -26210.4428263164
x53=−20277.3091634513x_{53} = -20277.3091634513
x54=−42314.7984614978x_{54} = -42314.7984614978
x55=39903.2310830382x_{55} = 39903.2310830382
x56=17865.7867262802x_{56} = 17865.7867262802
x57=−32991.2092898821x_{57} = -32991.2092898821
x58=−9258.95879064063x_{58} = -9258.95879064063
x59=−21972.4849200131x_{59} = -21972.4849200131
x60=−10954.0235214509x_{60} = -10954.0235214509
x61=29732.0539604744x_{61} = 29732.0539604744
x62=−11801.5713738084x_{62} = -11801.5713738084
x63=−21124.8963853792x_{63} = -21124.8963853792
x64=−16886.977356312x_{64} = -16886.977356312
x65=−22820.0746210752x_{65} = -22820.0746210752
x66=5152.91086370651x_{66} = 5152.91086370651
x67=31427.246834305x_{67} = 31427.246834305
x68=−13496.6883257698x_{68} = -13496.6883257698
x69=−7563.95850210681x_{69} = -7563.95850210681
x70=17018.2064590623x_{70} = 17018.2064590623
x71=9390.17878234122x_{71} = 9390.17878234122
x72=24646.4882132137x_{72} = 24646.4882132137
x73=22951.3055070382x_{73} = 22951.3055070382
x74=−33838.8069978398x_{74} = -33838.8069978398
x75=−24515.2570385241x_{75} = -24515.2570385241
x76=40750.8308374762x_{76} = 40750.8308374762
x77=−36381.6019467943x_{77} = -36381.6019467943
x78=−25362.8495533661x_{78} = -25362.8495533661
x79=6847.70613289343x_{79} = 6847.70613289343
x80=19560.9534923245x_{80} = 19560.9534923245
x81=38208.032170686x_{81} = 38208.032170686
x82=−27058.0367861288x_{82} = -27058.0367861288
x83=−16039.4000286848x_{83} = -16039.4000286848
x84=−19429.7234260341x_{84} = -19429.7234260341
x85=−37229.2008022157x_{85} = -37229.2008022157
x86=21256.1269105821x_{86} = 21256.1269105821
x87=−6716.49777568092x_{87} = -6716.49777568092
x88=11932.7963639546x_{88} = 11932.7963639546
x89=27189.2682966688x_{89} = 27189.2682966688
Signos de extremos en los puntos:
(36512.834145992565, 0.999999999943895)

(13627.915205912484, 0.999999999548784)

(-38924.399216070444, 0.999999999946964)

(-17734.55725617861, 0.99999999973503)

(16170.62870456547, 0.999999999682074)

(23798.896401298392, 0.999999999848202)

(41598.43077453214, 0.999999999943836)

(-40619.59846946623, 0.999999999949068)

(12780.352812712275, 0.999999999491137)

(26341.67423567186, 0.999999999878544)

(30579.650181215387, 0.999999999904241)

(10237.707205833769, 0.999999999204571)

(28884.458210140507, 0.999999999895273)

(39055.63152313194, 0.999999999939318)

(-28753.226523359415, 0.999999999898373)

(28036.86297287614, 0.999999999889254)

(-12649.126783205136, 0.999999999479564)

(-39771.998744535245, 0.999999999946935)

(-29600.822196685, 0.999999999903124)

(-41467.198378807785, 0.999999999951626)

(20408.53947353069, 0.999999999802226)

(33970.03906401565, 0.9999999999319)

(-30448.418346744576, 0.99999999990841)

(15323.053879770945, 0.99999999964837)

(18713.369164786425, 0.999999999762062)

(35665.23550608033, 0.999999999929527)

(14475.482499325431, 0.999999999600144)

(33122.44130475478, 0.999999999925651)

(22103.715635997123, 0.999999999830137)

(-32143.61192621353, 0.999999999918018)

(-23667.665363242217, 0.99999999984895)

(25494.080851254443, 0.999999999875735)

(-27905.631370212333, 0.99999999989415)

(-38076.79989719149, 0.999999999939754)

(-8411.448155553366, 0.999999998821564)

(-14344.254914200106, 0.999999999596252)

(-31296.014934803297, 0.999999999912904)

(-18582.139376279127, 0.999999999760366)

(37360.43303983254, 0.999999999943504)

(7695.172042907301, 0.999999998594119)

(32274.84388567656, 0.999999999921211)

(-5869.078512874751, 0.999999997580107)

(-10106.485127725944, 0.999999999183488)

(8542.665404040546, 0.999999998857507)

(-35534.003348062775, 0.999999999931898)

(34817.63713862056, 0.999999999929905)

(6000.279309089443, 0.999999997684569)

(11085.247223255352, 0.999999999323776)

(-15191.825703865761, 0.999999999637211)

(42446.03088328492, 0.999999999949988)

(-34686.405024843574, 0.99999999993407)

(-26210.44282631641, 0.9999999998789)

(-20277.309163451333, 0.999999999795962)

(-42314.79846149776, 0.99999999995506)

(39903.23108303823, 0.999999999954422)

(17865.786726280232, 0.999999999739458)

(-32991.209289882085, 0.999999999919875)

(-9258.958790640634, 0.99999999902765)

(-21972.484920013052, 0.999999999828907)

(-10954.023521450948, 0.999999999305978)

(29732.053960474437, 0.999999999902336)

(-11801.571373808376, 0.999999999400516)

(-21124.89638537924, 0.999999999814505)

(-16886.977356312047, 0.999999999708111)

(-22820.07462107522, 0.999999999840901)

(5152.910863706506, 0.999999996860707)

(31427.24683430499, 0.999999999909635)

(-13496.688325769803, 0.999999999542028)

(-7563.958502106809, 0.999999998544044)

(17018.2064590623, 0.999999999715423)

(9390.178782341221, 0.999999999056659)

(24646.48821321372, 0.999999999867149)

(22951.30550703823, 0.999999999842787)

(-33838.80699783976, 0.999999999926482)

(-24515.257038524076, 0.999999999859095)

(40750.83083747623, 0.99999999995989)

(-36381.60194679426, 0.999999999935505)

(-25362.849553366064, 0.999999999871507)

(6847.706132893425, 0.99999999822187)

(19560.953492324457, 0.999999999780259)

(38208.03217068598, 0.999999999951217)

(-27058.03678612882, 0.99999999988755)

(-16039.400028684786, 0.999999999677059)

(-19429.72342603411, 0.99999999977758)

(-37229.200802215695, 0.999999999936696)

(21256.126910582065, 0.999999999813405)

(-6716.49777568092, 0.999999998153448)

(11932.796363954556, 0.999999999412703)

(27189.26829666882, 0.999999999884665)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=13627.9152059125x_{1} = 13627.9152059125
x2=41598.4307745321x_{2} = 41598.4307745321
x3=−40619.5984694662x_{3} = -40619.5984694662
x4=26341.6742356719x_{4} = 26341.6742356719
x5=30579.6501812154x_{5} = 30579.6501812154
x6=28884.4582101405x_{6} = 28884.4582101405
x7=28036.8629728761x_{7} = 28036.8629728761
x8=35665.2355060803x_{8} = 35665.2355060803
x9=14475.4824993254x_{9} = 14475.4824993254
x10=−23667.6653632422x_{10} = -23667.6653632422
x11=−8411.44815555337x_{11} = -8411.44815555337
x12=−5869.07851287475x_{12} = -5869.07851287475
x13=−15191.8257038658x_{13} = -15191.8257038658
x14=42446.0308832849x_{14} = 42446.0308832849
x15=−32991.2092898821x_{15} = -32991.2092898821
x16=−11801.5713738084x_{16} = -11801.5713738084
x17=5152.91086370651x_{17} = 5152.91086370651
x18=31427.246834305x_{18} = 31427.246834305
x19=−13496.6883257698x_{19} = -13496.6883257698
x20=19560.9534923245x_{20} = 19560.9534923245
x21=−37229.2008022157x_{21} = -37229.2008022157
x22=11932.7963639546x_{22} = 11932.7963639546
Puntos máximos de la función:
x22=36512.8341459926x_{22} = 36512.8341459926
x22=12780.3528127123x_{22} = 12780.3528127123
x22=15323.0538797709x_{22} = 15323.0538797709
x22=−27905.6313702123x_{22} = -27905.6313702123
x22=−14344.2549142001x_{22} = -14344.2549142001
x22=−18582.1393762791x_{22} = -18582.1393762791
x22=11085.2472232554x_{22} = 11085.2472232554
x22=−34686.4050248436x_{22} = -34686.4050248436
x22=39903.2310830382x_{22} = 39903.2310830382
x22=−21972.4849200131x_{22} = -21972.4849200131
x22=−21124.8963853792x_{22} = -21124.8963853792
x22=−16886.977356312x_{22} = -16886.977356312
x22=−22820.0746210752x_{22} = -22820.0746210752
x22=−7563.95850210681x_{22} = -7563.95850210681
x22=17018.2064590623x_{22} = 17018.2064590623
x22=24646.4882132137x_{22} = 24646.4882132137
x22=22951.3055070382x_{22} = 22951.3055070382
x22=40750.8308374762x_{22} = 40750.8308374762
x22=38208.032170686x_{22} = 38208.032170686
x22=−27058.0367861288x_{22} = -27058.0367861288
x22=−6716.49777568092x_{22} = -6716.49777568092
Decrece en los intervalos
[42446.0308832849,∞)\left[42446.0308832849, \infty\right)
Crece en los intervalos
(−∞,−40619.5984694662]\left(-\infty, -40619.5984694662\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
(−2((e1x−1)2−2(e1x−1)e1xx+(e1x−1)e1xx2+e2xx2e1x−1−2(1−e1xx(e1x−1))(x(e1x−1)−e1x)x−2(x(e1x−1)−e1x)x+2(x(e1x−1)−e1x)e1xx2(e1x−1))e1x−1+2+1xx+4(x(e1x−1)−e1x)x2(e1x−1))e1xx4(e1x−1)2=0\frac{\left(- \frac{2 \left(\frac{\left(e^{\frac{1}{x}} - 1\right)^{2} - \frac{2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x} + \frac{\left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x^{2}} + \frac{e^{\frac{2}{x}}}{x^{2}}}{e^{\frac{1}{x}} - 1} - \frac{2 \left(1 - \frac{e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)}\right) \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} - \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} + \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right)}{e^{\frac{1}{x}} - 1} + \frac{2 + \frac{1}{x}}{x} + \frac{4 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=4378.45489343224x_{1} = 4378.45489343224
x2=−8051.80396529387x_{2} = -8051.80396529387
x3=6122.96954310366x_{3} = 6122.96954310366
x4=9612.05023180978x_{4} = 9612.05023180978
x5=2415.95737140673x_{5} = 2415.95737140673
x6=6559.10212778111x_{6} = 6559.10212778111
x7=8085.57286482282x_{7} = 8085.57286482282
x8=−9796.34976305168x_{8} = -9796.34976305168
x9=−5216.93993305863x_{9} = -5216.93993305863
x10=2634.00275086831x_{10} = 2634.00275086831
x11=3070.10409968592x_{11} = 3070.10409968592
x12=−4998.8757695151x_{12} = -4998.8757695151
x13=−7179.53399063609x_{13} = -7179.53399063609
x14=10484.3248752431x_{14} = 10484.3248752431
x15=10702.3936959841x_{15} = 10702.3936959841
x16=8957.84505151282x_{16} = 8957.84505151282
x17=−6961.46692563361x_{17} = -6961.46692563361
x18=5468.77302131714x_{18} = 5468.77302131714
x19=−2600.23715088493x_{19} = -2600.23715088493
x20=−4126.62552828055x_{20} = -4126.62552828055
x21=7431.37007033236x_{21} = 7431.37007033236
x22=−7615.66866449652x_{22} = -7615.66866449652
x23=−10450.5558183163x_{23} = -10450.5558183163
x24=2197.91692269672x_{24} = 2197.91692269672
x25=−6089.20093298288x_{25} = -6089.20093298288
x26=889.980807079997x_{26} = 889.980807079997
x27=−856.244713551305x_{27} = -856.244713551305
x28=10048.1874172931x_{28} = 10048.1874172931
x29=−1946.12030197999x_{29} = -1946.12030197999
x30=7649.43751843264x_{30} = 7649.43751843264
x31=−2382.19247235659x_{31} = -2382.19247235659
x32=7867.50512075532x_{32} = 7867.50512075532
x33=6995.23569448843x_{33} = 6995.23569448843
x34=5032.6440531647x_{34} = 5032.6440531647
x35=5250.70829861068x_{35} = 5250.70829861068
x36=−8269.87181979443x_{36} = -8269.87181979443
x37=10266.25611438x_{37} = 10266.25611438
x38=9175.91335764308x_{38} = 9175.91335764308
x39=−7397.60124226819x_{39} = -7397.60124226819
x40=−8706.00788587186x_{40} = -8706.00788587186
x41=9830.11878822658x_{41} = 9830.11878822658
x42=−1074.14251057279x_{42} = -1074.14251057279
x43=1107.89055578526x_{43} = 1107.89055578526
x44=1543.84605417316x_{44} = 1543.84605417316
x45=−8924.07607999643x_{45} = -8924.07607999643
x46=−10886.6934980213x_{46} = -10886.6934980213
x47=−6743.40006536826x_{47} = -6743.40006536826
x48=8303.64073946823x_{48} = 8303.64073946823
x49=−10232.4870673643x_{49} = -10232.4870673643
x50=6777.1688002299x_{50} = 6777.1688002299
x51=−2818.28578470162x_{51} = -2818.28578470162
x52=5686.83816638426x_{52} = 5686.83816638426
x53=8521.70873466762x_{53} = 8521.70873466762
x54=10920.4625730159x_{54} = 10920.4625730159
x55=−1728.09715000669x_{55} = -1728.09715000669
x56=4814.58034982471x_{56} = 4814.58034982471
x57=−4780.81215949822x_{57} = -4780.81215949822
x58=−9142.14437126442x_{58} = -9142.14437126442
x59=2852.05193017091x_{59} = 2852.05193017091
x60=4160.39334481353x_{60} = 4160.39334481353
x61=−6525.33343036833x_{61} = -6525.33343036833
x62=5904.90368702149x_{62} = 5904.90368702149
x63=−2164.15294523236x_{63} = -2164.15294523236
x64=−10014.4183808417x_{64} = -10014.4183808417
x65=6341.03570002487x_{65} = 6341.03570002487
x66=1979.88303384799x_{66} = 1979.88303384799
x67=1761.85814045997x_{67} = 1761.85814045997
x68=−3908.56510799633x_{68} = -3908.56510799633
x69=1325.85309427006x_{69} = 1325.85309427006
x70=−6307.26704400142x_{70} = -6307.26704400142
x71=−1292.09855768527x_{71} = -1292.09855768527
x72=3506.21517784186x_{72} = 3506.21517784186
x73=−5871.13512799763x_{73} = -5871.13512799763
x74=−3036.33752139337x_{74} = -3036.33752139337
x75=3724.27329735441x_{75} = 3724.27329735441
x76=−9578.28121868891x_{76} = -9578.28121868891
x77=9393.98175310191x_{77} = 9393.98175310191
x78=−3690.50584726659x_{78} = -3690.50584726659
x79=8739.77684139745x_{79} = 8739.77684139745
x80=−3472.44796459652x_{80} = -3472.44796459652
x81=−10668.6246297469x_{81} = -10668.6246297469
x82=−7833.73624307318x_{82} = -7833.73624307318
x83=4596.51726574094x_{83} = 4596.51726574094
x84=3288.15866454085x_{84} = 3288.15866454085
x85=−8487.93979637861x_{85} = -8487.93979637861
x86=−5435.00458349307x_{86} = -5435.00458349307
x87=−4344.68693348994x_{87} = -4344.68693348994
x88=−3254.39173707057x_{88} = -3254.39173707057
x89=7213.30279044085x_{89} = 7213.30279044085
x90=−1510.08760379866x_{90} = -1510.08760379866
x91=−9360.2127528858x_{91} = -9360.2127528858
x92=−5653.06966446584x_{92} = -5653.06966446584
x93=−4562.74918238466x_{93} = -4562.74918238466
x94=3942.33275655982x_{94} = 3942.33275655982
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−((−2((e1x−1)2−2(e1x−1)e1xx+(e1x−1)e1xx2+e2xx2e1x−1−2(1−e1xx(e1x−1))(x(e1x−1)−e1x)x−2(x(e1x−1)−e1x)x+2(x(e1x−1)−e1x)e1xx2(e1x−1))e1x−1+2+1xx+4(x(e1x−1)−e1x)x2(e1x−1))e1xx4(e1x−1)2)=0\lim_{x \to 0^-}\left(\frac{\left(- \frac{2 \left(\frac{\left(e^{\frac{1}{x}} - 1\right)^{2} - \frac{2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x} + \frac{\left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x^{2}} + \frac{e^{\frac{2}{x}}}{x^{2}}}{e^{\frac{1}{x}} - 1} - \frac{2 \left(1 - \frac{e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)}\right) \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} - \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} + \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right)}{e^{\frac{1}{x}} - 1} + \frac{2 + \frac{1}{x}}{x} + \frac{4 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = 0
lim⁡x→0+((−2((e1x−1)2−2(e1x−1)e1xx+(e1x−1)e1xx2+e2xx2e1x−1−2(1−e1xx(e1x−1))(x(e1x−1)−e1x)x−2(x(e1x−1)−e1x)x+2(x(e1x−1)−e1x)e1xx2(e1x−1))e1x−1+2+1xx+4(x(e1x−1)−e1x)x2(e1x−1))e1xx4(e1x−1)2)=−∞\lim_{x \to 0^+}\left(\frac{\left(- \frac{2 \left(\frac{\left(e^{\frac{1}{x}} - 1\right)^{2} - \frac{2 \left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x} + \frac{\left(e^{\frac{1}{x}} - 1\right) e^{\frac{1}{x}}}{x^{2}} + \frac{e^{\frac{2}{x}}}{x^{2}}}{e^{\frac{1}{x}} - 1} - \frac{2 \left(1 - \frac{e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)}\right) \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} - \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x} + \frac{2 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right) e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right)}{e^{\frac{1}{x}} - 1} + \frac{2 + \frac{1}{x}}{x} + \frac{4 \left(x \left(e^{\frac{1}{x}} - 1\right) - e^{\frac{1}{x}}\right)}{x^{2} \left(e^{\frac{1}{x}} - 1\right)}\right) e^{\frac{1}{x}}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = -\infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[10048.1874172931,∞)\left[10048.1874172931, \infty\right)
Convexa en los intervalos
(−∞,−10450.5558183163]\left(-\infty, -10450.5558183163\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(e1xx2(e1x−1)2)=1\lim_{x \to -\infty}\left(\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=1y = 1
lim⁡x→∞(e1xx2(e1x−1)2)=1\lim_{x \to \infty}\left(\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=1y = 1
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función exp(1/x)/((x^2*(exp(1/x) - 1)^2)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(1x2(e1x−1)2e1xx)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} e^{\frac{1}{x}}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(1x2(e1x−1)2e1xx)=0\lim_{x \to \infty}\left(\frac{\frac{1}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} e^{\frac{1}{x}}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
e1xx2(e1x−1)2=e−1xx2(−1+e−1x)2\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} = \frac{e^{- \frac{1}{x}}}{x^{2} \left(-1 + e^{- \frac{1}{x}}\right)^{2}}
- No
e1xx2(e1x−1)2=−e−1xx2(−1+e−1x)2\frac{e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}} = - \frac{e^{- \frac{1}{x}}}{x^{2} \left(-1 + e^{- \frac{1}{x}}\right)^{2}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор