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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       sin(x)
f(x) = ------
            2
       1 - x
f(x)=sin⁡(x)1−x2f{\left(x \right)} = \frac{\sin{\left(x \right)}}{1 - x^{2}} 
f = sin(x)/(1 - x^2)
Gráfico de la función
02468-8-6-4-2-1010-2020
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−1x_{1} = -1
x2=1x_{2} = 1
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:
sin⁡(x)1−x2=0\frac{\sin{\left(x \right)}}{1 - x^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=31.4159265358979x_{1} = 31.4159265358979
x2=3.14159265358979x_{2} = 3.14159265358979
x3=−414.690230273853x_{3} = -414.690230273853
x4=−47.1238898038469x_{4} = -47.1238898038469
x5=−12.5663706143592x_{5} = -12.5663706143592
x6=−34.5575191894877x_{6} = -34.5575191894877
x7=−69.1150383789755x_{7} = -69.1150383789755
x8=75.398223686155x_{8} = 75.398223686155
x9=−65.9734457253857x_{9} = -65.9734457253857
x10=−50.2654824574367x_{10} = -50.2654824574367
x11=−56.5486677646163x_{11} = -56.5486677646163
x12=59.6902604182061x_{12} = 59.6902604182061
x13=72.2566310325652x_{13} = 72.2566310325652
x14=91.106186954104x_{14} = 91.106186954104
x15=−91.106186954104x_{15} = -91.106186954104
x16=−62.8318530717959x_{16} = -62.8318530717959
x17=−6.28318530717959x_{17} = -6.28318530717959
x18=6.28318530717959x_{18} = 6.28318530717959
x19=62.8318530717959x_{19} = 62.8318530717959
x20=−25.1327412287183x_{20} = -25.1327412287183
x21=94.2477796076938x_{21} = 94.2477796076938
x22=−9.42477796076938x_{22} = -9.42477796076938
x23=−37.6991118430775x_{23} = -37.6991118430775
x24=65.9734457253857x_{24} = 65.9734457253857
x25=−100.530964914873x_{25} = -100.530964914873
x26=−43.9822971502571x_{26} = -43.9822971502571
x27=25.1327412287183x_{27} = 25.1327412287183
x28=21.9911485751286x_{28} = 21.9911485751286
x29=87.9645943005142x_{29} = 87.9645943005142
x30=−40.8407044966673x_{30} = -40.8407044966673
x31=−97.3893722612836x_{31} = -97.3893722612836
x32=43.9822971502571x_{32} = 43.9822971502571
x33=−53.4070751110265x_{33} = -53.4070751110265
x34=97.3893722612836x_{34} = 97.3893722612836
x35=100.530964914873x_{35} = 100.530964914873
x36=−94.2477796076938x_{36} = -94.2477796076938
x37=−31.4159265358979x_{37} = -31.4159265358979
x38=18.8495559215388x_{38} = 18.8495559215388
x39=78.5398163397448x_{39} = 78.5398163397448
x40=−18.8495559215388x_{40} = -18.8495559215388
x41=53.4070751110265x_{41} = 53.4070751110265
x42=47.1238898038469x_{42} = 47.1238898038469
x43=12.5663706143592x_{43} = 12.5663706143592
x44=81.6814089933346x_{44} = 81.6814089933346
x45=34.5575191894877x_{45} = 34.5575191894877
x46=−75.398223686155x_{46} = -75.398223686155
x47=−144.51326206513x_{47} = -144.51326206513
x48=−15.707963267949x_{48} = -15.707963267949
x49=50.2654824574367x_{49} = 50.2654824574367
x50=−81.6814089933346x_{50} = -81.6814089933346
x51=248.185819633594x_{51} = 248.185819633594
x52=−3.14159265358979x_{52} = -3.14159265358979
x53=−59.6902604182061x_{53} = -59.6902604182061
x54=−28.2743338823081x_{54} = -28.2743338823081
x55=−87.9645943005142x_{55} = -87.9645943005142
x56=9.42477796076938x_{56} = 9.42477796076938
x57=−21.9911485751286x_{57} = -21.9911485751286
x58=−113.097335529233x_{58} = -113.097335529233
x59=56.5486677646163x_{59} = 56.5486677646163
x60=15.707963267949x_{60} = 15.707963267949
x61=84.8230016469244x_{61} = 84.8230016469244
x62=−78.5398163397448x_{62} = -78.5398163397448
x63=37.6991118430775x_{63} = 37.6991118430775
x64=−72.2566310325652x_{64} = -72.2566310325652
x65=−84.8230016469244x_{65} = -84.8230016469244
x66=69.1150383789755x_{66} = 69.1150383789755
x67=0x_{67} = 0
x68=28.2743338823081x_{68} = 28.2743338823081
x69=40.8407044966673x_{69} = 40.8407044966673
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 sin(x)/(1 - x^2).
sin⁡(0)1−02\frac{\sin{\left(0 \right)}}{1 - 0^{2}}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
2xsin⁡(x)(1−x2)2+cos⁡(x)1−x2=0\frac{2 x \sin{\left(x \right)}}{\left(1 - x^{2}\right)^{2}} + \frac{\cos{\left(x \right)}}{1 - x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=48.6535849776189x_{1} = 48.6535849776189
x2=−95.7976993646524x_{2} = -95.7976993646524
x3=−54.9414730837878x_{3} = -54.9414730837878
x4=−39.2189234266452x_{4} = -39.2189234266452
x5=−89.5130484454873x_{5} = -89.5130484454873
x6=26.6284652377851x_{6} = 26.6284652377851
x7=−7.59205618191083x_{7} = -7.59205618191083
x8=13.9944961126907x_{8} = 13.9944961126907
x9=76.9430282181184x_{9} = 76.9430282181184
x10=10.8111042087213x_{10} = 10.8111042087213
x11=−23.4768059032848x_{11} = -23.4768059032848
x12=39.2189234266452x_{12} = 39.2189234266452
x13=−36.0728864084812x_{13} = -36.0728864084812
x14=−32.9259992567895x_{14} = -32.9259992567895
x15=−61.2283950657729x_{15} = -61.2283950657729
x16=29.7779917432681x_{16} = 29.7779917432681
x17=80.0856406984281x_{17} = 80.0856406984281
x18=83.2281761528687x_{18} = 83.2281761528687
x19=−29.7779917432681x_{19} = -29.7779917432681
x20=−42.3643000278463x_{20} = -42.3643000278463
x21=−10.8111042087213x_{21} = -10.8111042087213
x22=−13.9944961126907x_{22} = -13.9944961126907
x23=−26.6284652377851x_{23} = -26.6284652377851
x24=89.5130484454873x_{24} = 89.5130484454873
x25=70.6575310493539x_{25} = 70.6575310493539
x26=−120.934779700424x_{26} = -120.934779700424
x27=−86.3706429922226x_{27} = -86.3706429922226
x28=92.6553987604331x_{28} = 92.6553987604331
x29=23.4768059032848x_{29} = 23.4768059032848
x30=67.5146210051587x_{30} = 67.5146210051587
x31=136.644644187573x_{31} = 136.644644187573
x32=61.2283950657729x_{32} = 61.2283950657729
x33=−4.2502319840436x_{33} = -4.2502319840436
x34=20.3220161353369x_{34} = 20.3220161353369
x35=98.9399549958912x_{35} = 98.9399549958912
x36=−67.5146210051587x_{36} = -67.5146210051587
x37=73.8003288675086x_{37} = 73.8003288675086
x38=45.5091533451563x_{38} = 45.5091533451563
x39=17.1623570970183x_{39} = 17.1623570970183
x40=95.7976993646524x_{40} = 95.7976993646524
x41=−45.5091533451563x_{41} = -45.5091533451563
x42=−212.048072363693x_{42} = -212.048072363693
x43=−73.8003288675086x_{43} = -73.8003288675086
x44=−48.6535849776189x_{44} = -48.6535849776189
x45=54.9414730837878x_{45} = 54.9414730837878
x46=4.2502319840436x_{46} = 4.2502319840436
x47=64.3715822869017x_{47} = 64.3715822869017
x48=86.3706429922226x_{48} = 86.3706429922226
x49=36.0728864084812x_{49} = 36.0728864084812
x50=58.0850352160434x_{50} = 58.0850352160434
x51=−92.6553987604331x_{51} = -92.6553987604331
x52=51.7976718062027x_{52} = 51.7976718062027
x53=−98.9399549958912x_{53} = -98.9399549958912
x54=−20.3220161353369x_{54} = -20.3220161353369
x55=−83.2281761528687x_{55} = -83.2281761528687
x56=42.3643000278463x_{56} = 42.3643000278463
x57=−58.0850352160434x_{57} = -58.0850352160434
x58=−64.3715822869017x_{58} = -64.3715822869017
x59=−70.6575310493539x_{59} = -70.6575310493539
x60=−17.1623570970183x_{60} = -17.1623570970183
x61=−76.9430282181184x_{61} = -76.9430282181184
x62=7.59205618191083x_{62} = 7.59205618191083
x63=−51.7976718062027x_{63} = -51.7976718062027
x64=472.805464302016x_{64} = 472.805464302016
x65=32.9259992567895x_{65} = 32.9259992567895
x66=−80.0856406984281x_{66} = -80.0856406984281
Signos de extremos en los puntos:
(48.653584977618934, 0.000422266745161591)

(-95.79769936465237, 0.000108953834823294)

(-54.941473083787834, -0.00033117347900486)

(-39.2189234266452, 0.000649720249572517)

(-89.5130484454873, 0.000124788081105762)

(26.62846523778512, -0.00140830162078119)

(-7.592056181910829, 0.0170534046093743)

(13.994496112690735, -0.00508011541536346)

(76.94302821811836, -0.000168883841534665)

(10.811104208721284, 0.00848320511338927)

(-23.476805903284752, -0.00181106789272178)

(39.2189234266452, -0.000649720249572517)

(-36.072886408481175, -0.000767899860645288)

(-32.925999256789495, 0.00092155585086097)

(-61.22839506577286, -0.000266672614366939)

(29.777991743268093, 0.0011264701660197)

(80.0856406984281, 0.000155891696428242)

(83.22817615286866, -0.000144343274284351)

(-29.777991743268093, -0.0011264701660197)

(-42.36430002784626, -0.000556875375921231)

(-10.811104208721284, -0.00848320511338927)

(-13.994496112690735, 0.00508011541536346)

(-26.62846523778512, 0.00140830162078119)

(89.5130484454873, -0.000124788081105762)

(70.65753104935389, -0.000200260872068306)

(-120.93477970042365, 6.83703606024318e-5)

(-86.37064299222264, -0.000134032303380199)

(92.65539876043312, 0.000116468359027335)

(23.476805903284752, 0.00181106789272178)

(67.51462100515869, 0.000219335568761136)

(136.6446441875733, 5.35539505172292e-5)

(61.22839506577286, 0.000266672614366939)

(-4.250231984043597, -0.0524535903376383)

(20.322016135336863, -0.0024155503091525)

(98.93995499589117, 0.000102143849300632)

(-67.51462100515869, -0.000219335568761136)

(73.80032886750858, 0.000183570831307369)

(45.509153345156335, -0.000482606141533831)

(17.162357097018344, 0.00338356230302691)

(95.79769936465237, -0.000108953834823294)

(-45.509153345156335, 0.000482606141533831)

(-212.0480723636933, -2.22393292118406e-5)

(-73.80032886750858, -0.000183570831307369)

(-48.653584977618934, -0.000422266745161591)

(54.941473083787834, 0.00033117347900486)

(4.250231984043597, 0.0524535903376383)

(64.37158228690171, -0.000241271950626197)

(86.37064299222264, 0.000134032303380199)

(36.072886408481175, 0.000767899860645288)

(58.08503521604338, -0.000296307594083531)

(-92.65539876043312, -0.000116468359027335)

(51.79767180620269, -0.000372578407377583)

(-98.93995499589117, -0.000102143849300632)

(-20.322016135336863, 0.0024155503091525)

(-83.22817615286866, 0.000144343274284351)

(42.36430002784626, 0.000556875375921231)

(-58.08503521604338, 0.000296307594083531)

(-64.37158228690171, 0.000241271950626197)

(-70.65753104935389, 0.000200260872068306)

(-17.162357097018344, -0.00338356230302691)

(-76.94302821811836, 0.000168883841534665)

(7.592056181910829, -0.0170534046093743)

(-51.79767180620269, 0.000372578407377583)

(472.8054643020164, -4.47335209914827e-6)

(32.925999256789495, -0.00092155585086097)

(-80.0856406984281, -0.000155891696428242)


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=−54.9414730837878x_{1} = -54.9414730837878
x2=26.6284652377851x_{2} = 26.6284652377851
x3=13.9944961126907x_{3} = 13.9944961126907
x4=76.9430282181184x_{4} = 76.9430282181184
x5=−23.4768059032848x_{5} = -23.4768059032848
x6=39.2189234266452x_{6} = 39.2189234266452
x7=−36.0728864084812x_{7} = -36.0728864084812
x8=−61.2283950657729x_{8} = -61.2283950657729
x9=83.2281761528687x_{9} = 83.2281761528687
x10=−29.7779917432681x_{10} = -29.7779917432681
x11=−42.3643000278463x_{11} = -42.3643000278463
x12=−10.8111042087213x_{12} = -10.8111042087213
x13=89.5130484454873x_{13} = 89.5130484454873
x14=70.6575310493539x_{14} = 70.6575310493539
x15=−86.3706429922226x_{15} = -86.3706429922226
x16=−4.2502319840436x_{16} = -4.2502319840436
x17=20.3220161353369x_{17} = 20.3220161353369
x18=−67.5146210051587x_{18} = -67.5146210051587
x19=45.5091533451563x_{19} = 45.5091533451563
x20=95.7976993646524x_{20} = 95.7976993646524
x21=−212.048072363693x_{21} = -212.048072363693
x22=−73.8003288675086x_{22} = -73.8003288675086
x23=−48.6535849776189x_{23} = -48.6535849776189
x24=64.3715822869017x_{24} = 64.3715822869017
x25=58.0850352160434x_{25} = 58.0850352160434
x26=−92.6553987604331x_{26} = -92.6553987604331
x27=51.7976718062027x_{27} = 51.7976718062027
x28=−98.9399549958912x_{28} = -98.9399549958912
x29=−17.1623570970183x_{29} = -17.1623570970183
x30=7.59205618191083x_{30} = 7.59205618191083
x31=472.805464302016x_{31} = 472.805464302016
x32=32.9259992567895x_{32} = 32.9259992567895
x33=−80.0856406984281x_{33} = -80.0856406984281
Puntos máximos de la función:
x33=48.6535849776189x_{33} = 48.6535849776189
x33=−95.7976993646524x_{33} = -95.7976993646524
x33=−39.2189234266452x_{33} = -39.2189234266452
x33=−89.5130484454873x_{33} = -89.5130484454873
x33=−7.59205618191083x_{33} = -7.59205618191083
x33=10.8111042087213x_{33} = 10.8111042087213
x33=−32.9259992567895x_{33} = -32.9259992567895
x33=29.7779917432681x_{33} = 29.7779917432681
x33=80.0856406984281x_{33} = 80.0856406984281
x33=−13.9944961126907x_{33} = -13.9944961126907
x33=−26.6284652377851x_{33} = -26.6284652377851
x33=−120.934779700424x_{33} = -120.934779700424
x33=92.6553987604331x_{33} = 92.6553987604331
x33=23.4768059032848x_{33} = 23.4768059032848
x33=67.5146210051587x_{33} = 67.5146210051587
x33=136.644644187573x_{33} = 136.644644187573
x33=61.2283950657729x_{33} = 61.2283950657729
x33=98.9399549958912x_{33} = 98.9399549958912
x33=73.8003288675086x_{33} = 73.8003288675086
x33=17.1623570970183x_{33} = 17.1623570970183
x33=−45.5091533451563x_{33} = -45.5091533451563
x33=54.9414730837878x_{33} = 54.9414730837878
x33=4.2502319840436x_{33} = 4.2502319840436
x33=86.3706429922226x_{33} = 86.3706429922226
x33=36.0728864084812x_{33} = 36.0728864084812
x33=−20.3220161353369x_{33} = -20.3220161353369
x33=−83.2281761528687x_{33} = -83.2281761528687
x33=42.3643000278463x_{33} = 42.3643000278463
x33=−58.0850352160434x_{33} = -58.0850352160434
x33=−64.3715822869017x_{33} = -64.3715822869017
x33=−70.6575310493539x_{33} = -70.6575310493539
x33=−76.9430282181184x_{33} = -76.9430282181184
x33=−51.7976718062027x_{33} = -51.7976718062027
Decrece en los intervalos
[472.805464302016,∞)\left[472.805464302016, \infty\right)
Crece en los intervalos
(−∞,−212.048072363693]\left(-\infty, -212.048072363693\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
4xcos⁡(x)x2−1+sin⁡(x)−2(4x2x2−1−1)sin⁡(x)x2−1x2−1=0\frac{\frac{4 x \cos{\left(x \right)}}{x^{2} - 1} + \sin{\left(x \right)} - \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \sin{\left(x \right)}}{x^{2} - 1}}{x^{2} - 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=37.5925825893023x_{1} = 37.5925825893023
x2=−8.9698877316131x_{2} = -8.9698877316131
x3=−5.5229319930101x_{3} = -5.5229319930101
x4=72.2012125964132x_{4} = 72.2012125964132
x5=−50.1857258235068x_{5} = -50.1857258235068
x6=−15.4472262173652x_{6} = -15.4472262173652
x7=−91.0622521332484x_{7} = -91.0622521332484
x8=−84.7758074362733x_{8} = -84.7758074362733
x9=87.9190881163862x_{9} = 87.9190881163862
x10=−28.1318477063748x_{10} = -28.1318477063748
x11=43.8910837148979x_{11} = 43.8910837148979
x12=−53.3320293640533x_{12} = -53.3320293640533
x13=53.3320293640533x_{13} = 53.3320293640533
x14=78.4888398981535x_{14} = 78.4888398981535
x15=−34.4412164059948x_{15} = -34.4412164059948
x16=−97.3482754540447x_{16} = -97.3482754540447
x17=−103.633954191491x_{17} = -103.633954191491
x18=94.2053111846222x_{18} = 94.2053111846222
x19=84.7758074362733x_{19} = 84.7758074362733
x20=−18.6338695613689x_{20} = -18.6338695613689
x21=−37.5925825893023x_{21} = -37.5925825893023
x22=−21.8070821937415x_{22} = -21.8070821937415
x23=18.6338695613689x_{23} = 18.6338695613689
x24=12.2358820049333x_{24} = 12.2358820049333
x25=56.4778065045334x_{25} = 56.4778065045334
x26=75.3451190666404x_{26} = 75.3451190666404
x27=100.491153848292x_{27} = 100.491153848292
x28=−65.91273615789x_{28} = -65.91273615789
x29=50.1857258235068x_{29} = 50.1857258235068
x30=−24.9721361872599x_{30} = -24.9721361872599
x31=15.4472262173652x_{31} = 15.4472262173652
x32=24.9721361872599x_{32} = 24.9721361872599
x33=0x_{33} = 0
x34=5.5229319930101x_{34} = 5.5229319930101
x35=81.632396588364x_{35} = 81.632396588364
x36=−31.2878642896602x_{36} = -31.2878642896602
x37=91.0622521332484x_{37} = 91.0622521332484
x38=−81.632396588364x_{38} = -81.632396588364
x39=21.8070821937415x_{39} = 21.8070821937415
x40=−78.4888398981535x_{40} = -78.4888398981535
x41=−69.0570950600626x_{41} = -69.0570950600626
x42=97.3482754540447x_{42} = 97.3482754540447
x43=241.886097146371x_{43} = 241.886097146371
x44=40.742428305889x_{44} = 40.742428305889
x45=65.91273615789x_{45} = 65.91273615789
x46=119.347001200998x_{46} = 119.347001200998
x47=−62.7680994904373x_{47} = -62.7680994904373
x48=−87.9190881163862x_{48} = -87.9190881163862
x49=−113.061952083617x_{49} = -113.061952083617
x50=−43.8910837148979x_{50} = -43.8910837148979
x51=−72.2012125964132x_{51} = -72.2012125964132
x52=59.6231409396343x_{52} = 59.6231409396343
x53=−40.742428305889x_{53} = -40.742428305889
x54=69.0570950600626x_{54} = 69.0570950600626
x55=47.03878961526x_{55} = 47.03878961526
x56=−94.2053111846222x_{56} = -94.2053111846222
x57=34.4412164059948x_{57} = 34.4412164059948
x58=−47.03878961526x_{58} = -47.03878961526
x59=−75.3451190666404x_{59} = -75.3451190666404
x60=8.9698877316131x_{60} = 8.9698877316131
x61=62.7680994904373x_{61} = 62.7680994904373
x62=−12.2358820049333x_{62} = -12.2358820049333
x63=−100.491153848292x_{63} = -100.491153848292
x64=28.1318477063748x_{64} = 28.1318477063748
x65=31.2878642896602x_{65} = 31.2878642896602
x66=−56.4778065045334x_{66} = -56.4778065045334
x67=−59.6231409396343x_{67} = -59.6231409396343
x68=109.919347538894x_{68} = 109.919347538894
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=−1x_{1} = -1
x2=1x_{2} = 1

lim⁡x→−1−(4xcos⁡(x)x2−1+sin⁡(x)−2(4x2x2−1−1)sin⁡(x)x2−1x2−1)=∞\lim_{x \to -1^-}\left(\frac{\frac{4 x \cos{\left(x \right)}}{x^{2} - 1} + \sin{\left(x \right)} - \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \sin{\left(x \right)}}{x^{2} - 1}}{x^{2} - 1}\right) = \infty
lim⁡x→−1+(4xcos⁡(x)x2−1+sin⁡(x)−2(4x2x2−1−1)sin⁡(x)x2−1x2−1)=−∞\lim_{x \to -1^+}\left(\frac{\frac{4 x \cos{\left(x \right)}}{x^{2} - 1} + \sin{\left(x \right)} - \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \sin{\left(x \right)}}{x^{2} - 1}}{x^{2} - 1}\right) = -\infty
- los límites no son iguales, signo
x1=−1x_{1} = -1
- es el punto de flexión
lim⁡x→1−(4xcos⁡(x)x2−1+sin⁡(x)−2(4x2x2−1−1)sin⁡(x)x2−1x2−1)=∞\lim_{x \to 1^-}\left(\frac{\frac{4 x \cos{\left(x \right)}}{x^{2} - 1} + \sin{\left(x \right)} - \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \sin{\left(x \right)}}{x^{2} - 1}}{x^{2} - 1}\right) = \infty
lim⁡x→1+(4xcos⁡(x)x2−1+sin⁡(x)−2(4x2x2−1−1)sin⁡(x)x2−1x2−1)=−∞\lim_{x \to 1^+}\left(\frac{\frac{4 x \cos{\left(x \right)}}{x^{2} - 1} + \sin{\left(x \right)} - \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \sin{\left(x \right)}}{x^{2} - 1}}{x^{2} - 1}\right) = -\infty
- los límites no son iguales, signo
x2=1x_{2} = 1
- 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
[119.347001200998,∞)\left[119.347001200998, \infty\right)
Convexa en los intervalos
(−∞,−113.061952083617]\left(-\infty, -113.061952083617\right]
Asíntotas verticales
Hay:
x1=−1x_{1} = -1
x2=1x_{2} = 1
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(x)1−x2)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{1 - x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(x)1−x2)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{1 - x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)/(1 - x^2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x)x(1−x2))=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - x^{2}\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(x)x(1−x2))=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - x^{2}\right)}\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:
sin⁡(x)1−x2=−sin⁡(x)1−x2\frac{\sin{\left(x \right)}}{1 - x^{2}} = - \frac{\sin{\left(x \right)}}{1 - x^{2}}
- No
sin⁡(x)1−x2=sin⁡(x)1−x2\frac{\sin{\left(x \right)}}{1 - x^{2}} = \frac{\sin{\left(x \right)}}{1 - x^{2}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор