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Trazado el gráfico de la función/ sinx/(x^2-4)

Gráfico de la función y = sinx/(x^2-4)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

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

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=−18.8495559215388x_{1} = -18.8495559215388
x2=−53.4070751110265x_{2} = -53.4070751110265
x3=−37.6991118430775x_{3} = -37.6991118430775
x4=128.805298797182x_{4} = 128.805298797182
x5=−15.707963267949x_{5} = -15.707963267949
x6=−59.6902604182061x_{6} = -59.6902604182061
x7=−122.522113490002x_{7} = -122.522113490002
x8=−56.5486677646163x_{8} = -56.5486677646163
x9=12.5663706143592x_{9} = 12.5663706143592
x10=−81.6814089933346x_{10} = -81.6814089933346
x11=−31.4159265358979x_{11} = -31.4159265358979
x12=84.8230016469244x_{12} = 84.8230016469244
x13=94.2477796076938x_{13} = 94.2477796076938
x14=21.9911485751286x_{14} = 21.9911485751286
x15=0x_{15} = 0
x16=−87.9645943005142x_{16} = -87.9645943005142
x17=81.6814089933346x_{17} = 81.6814089933346
x18=40.8407044966673x_{18} = 40.8407044966673
x19=−75.398223686155x_{19} = -75.398223686155
x20=−78.5398163397448x_{20} = -78.5398163397448
x21=62.8318530717959x_{21} = 62.8318530717959
x22=100.530964914873x_{22} = 100.530964914873
x23=−21.9911485751286x_{23} = -21.9911485751286
x24=47.1238898038469x_{24} = 47.1238898038469
x25=91.106186954104x_{25} = 91.106186954104
x26=75.398223686155x_{26} = 75.398223686155
x27=28.2743338823081x_{27} = 28.2743338823081
x28=−292.168116783851x_{28} = -292.168116783851
x29=34.5575191894877x_{29} = 34.5575191894877
x30=6.28318530717959x_{30} = 6.28318530717959
x31=78.5398163397448x_{31} = 78.5398163397448
x32=72.2566310325652x_{32} = 72.2566310325652
x33=−6.28318530717959x_{33} = -6.28318530717959
x34=15.707963267949x_{34} = 15.707963267949
x35=31.4159265358979x_{35} = 31.4159265358979
x36=−47.1238898038469x_{36} = -47.1238898038469
x37=25.1327412287183x_{37} = 25.1327412287183
x38=18.8495559215388x_{38} = 18.8495559215388
x39=−94.2477796076938x_{39} = -94.2477796076938
x40=−40.8407044966673x_{40} = -40.8407044966673
x41=56.5486677646163x_{41} = 56.5486677646163
x42=−25.1327412287183x_{42} = -25.1327412287183
x43=53.4070751110265x_{43} = 53.4070751110265
x44=−28.2743338823081x_{44} = -28.2743338823081
x45=−9.42477796076938x_{45} = -9.42477796076938
x46=87.9645943005142x_{46} = 87.9645943005142
x47=−50.2654824574367x_{47} = -50.2654824574367
x48=−100.530964914873x_{48} = -100.530964914873
x49=−43.9822971502571x_{49} = -43.9822971502571
x50=50.2654824574367x_{50} = 50.2654824574367
x51=−97.3893722612836x_{51} = -97.3893722612836
x52=69.1150383789755x_{52} = 69.1150383789755
x53=59.6902604182061x_{53} = 59.6902604182061
x54=97.3893722612836x_{54} = 97.3893722612836
x55=−62.8318530717959x_{55} = -62.8318530717959
x56=−72.2566310325652x_{56} = -72.2566310325652
x57=109.955742875643x_{57} = 109.955742875643
x58=−91.106186954104x_{58} = -91.106186954104
x59=−12.5663706143592x_{59} = -12.5663706143592
x60=−69.1150383789755x_{60} = -69.1150383789755
x61=37.6991118430775x_{61} = 37.6991118430775
x62=9.42477796076938x_{62} = 9.42477796076938
x63=65.9734457253857x_{63} = 65.9734457253857
x64=−65.9734457253857x_{64} = -65.9734457253857
x65=−84.8230016469244x_{65} = -84.8230016469244
x66=−34.5575191894877x_{66} = -34.5575191894877
x67=131.946891450771x_{67} = 131.946891450771
x68=43.9822971502571x_{68} = 43.9822971502571
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)/(x^2 - 4).
sin⁡(0)−4+02\frac{\sin{\left(0 \right)}}{-4 + 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)(x2−4)2+cos⁡(x)x2−4=0- \frac{2 x \sin{\left(x \right)}}{\left(x^{2} - 4\right)^{2}} + \frac{\cos{\left(x \right)}}{x^{2} - 4} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=45.5090895943713x_{1} = 45.5090895943713
x2=−80.0856290117635x_{2} = -80.0856290117635
x3=23.4763396531088x_{3} = 23.4763396531088
x4=4.15222477298826x_{4} = 4.15222477298826
x5=64.3715597764874x_{5} = 64.3715597764874
x6=70.657514030234x_{6} = 70.657514030234
x7=−23.4763396531088x_{7} = -23.4763396531088
x8=80.0856290117635x_{8} = 80.0856290117635
x9=−10.8062270456027x_{9} = -10.8062270456027
x10=−42.3642209820001x_{10} = -42.3642209820001
x11=−58.0850045720611x_{11} = -58.0850045720611
x12=73.8003139321454x_{12} = 73.8003139321454
x13=89.5130400768468x_{13} = 89.5130400768468
x14=−26.6281461112737x_{14} = -26.6281461112737
x15=−98.9399487990654x_{15} = -98.9399487990654
x16=−13.9922722908994x_{16} = -13.9922722908994
x17=−83.2281657410082x_{17} = -83.2281657410082
x18=67.51460149576x_{18} = 67.51460149576
x19=−67.51460149576x_{19} = -67.51460149576
x20=17.1611578256127x_{20} = 17.1611578256127
x21=−64.3715597764874x_{21} = -64.3715597764874
x22=−48.6535328151386x_{22} = -48.6535328151386
x23=−4.15222477298826x_{23} = -4.15222477298826
x24=−45.5090895943713x_{24} = -45.5090895943713
x25=61.2283689056001x_{25} = 61.2283689056001
x26=95.7976925376814x_{26} = 95.7976925376814
x27=−39.2188237682302x_{27} = -39.2188237682302
x28=−290.590437700776x_{28} = -290.590437700776
x29=42.3642209820001x_{29} = 42.3642209820001
x30=−7.57754404149509x_{30} = -7.57754404149509
x31=7.57754404149509x_{31} = 7.57754404149509
x32=−20.3212959521274x_{32} = -20.3212959521274
x33=−86.3706336762639x_{33} = -86.3706336762639
x34=83.2281657410082x_{34} = 83.2281657410082
x35=58.0850045720611x_{35} = 58.0850045720611
x36=−124.076787974095x_{36} = -124.076787974095
x37=54.9414368692733x_{37} = 54.9414368692733
x38=48.6535328151386x_{38} = 48.6535328151386
x39=−95.7976925376814x_{39} = -95.7976925376814
x40=−36.0727582894777x_{40} = -36.0727582894777
x41=10.8062270456027x_{41} = 10.8062270456027
x42=32.9258307018341x_{42} = 32.9258307018341
x43=−32.9258307018341x_{43} = -32.9258307018341
x44=98.9399487990654x_{44} = 98.9399487990654
x45=−17.1611578256127x_{45} = -17.1611578256127
x46=202.62285496058x_{46} = 202.62285496058
x47=36.0727582894777x_{47} = 36.0727582894777
x48=92.655391214883x_{48} = 92.655391214883
x49=−73.8003139321454x_{49} = -73.8003139321454
x50=−29.777763739304x_{50} = -29.777763739304
x51=−70.657514030234x_{51} = -70.657514030234
x52=−155.495972864649x_{52} = -155.495972864649
x53=26.6281461112737x_{53} = 26.6281461112737
x54=76.9430150396882x_{54} = 76.9430150396882
x55=−76.9430150396882x_{55} = -76.9430150396882
x56=−51.7976285839881x_{56} = -51.7976285839881
x57=20.3212959521274x_{57} = 20.3212959521274
x58=−61.2283689056001x_{58} = -61.2283689056001
x59=86.3706336762639x_{59} = 86.3706336762639
x60=−54.9414368692733x_{60} = -54.9414368692733
x61=−92.655391214883x_{61} = -92.655391214883
x62=51.7976285839881x_{62} = 51.7976285839881
x63=13.9922722908994x_{63} = 13.9922722908994
x64=−89.5130400768468x_{64} = -89.5130400768468
x65=39.2188237682302x_{65} = 39.2188237682302
x66=29.777763739304x_{66} = 29.777763739304
Signos de extremos en los puntos:
(45.50908959437133, 0.000483306558706209)

(-80.08562901176347, 0.000155964659973988)

(23.476339653108777, -0.00182099790998716)

(4.152224772988261, -0.0639807988925772)

(64.37155977648736, 0.000241446798078732)

(70.65751403023398, 0.000200381305938913)

(-23.476339653108777, 0.00182099790998716)

(80.08562901176347, -0.000155964659973988)

(-10.806227045602704, 0.0087087677703681)

(-42.36422098200009, 0.000557808305842402)

(-58.08500457206114, -0.000296571379601852)

(73.80031393214543, -0.000183672018953665)

(89.51304007684683, 0.000124834826472013)

(-26.62814611127368, -0.00141429382702696)

(-98.9399487990654, 0.000102175165394175)

(-13.992272290899354, -0.00515956885743496)

(-83.22816574100821, -0.000144405824378095)

(67.51460149576005, -0.000219480051529099)

(-67.51460149576005, 0.000219480051529099)

(17.16115782561268, -0.00341850130927523)

(-64.37155977648736, -0.000241446798078732)

(-48.653532815138554, 0.000422802805149809)

(-4.152224772988261, 0.0639807988925772)

(-45.50908959437133, -0.000483306558706209)

(61.22836890560009, -0.000266886242151348)

(95.7976925376814, 0.000108989467051541)

(-39.21882376823015, -0.000650990789484618)

(-290.5904377007758, -1.18426160073303e-5)

(42.36422098200009, -0.000557808305842402)

(-7.577544041495088, -0.0180091484531043)

(7.577544041495088, 0.0180091484531043)

(-20.32129595212744, -0.00243326959458963)

(-86.37063367626389, 0.000134086233502956)

(83.22816574100821, 0.000144405824378095)

(58.08500457206114, 0.000296571379601852)

(-124.07678797409463, 6.49643845271334e-5)

(54.94143686927333, -0.000331503052571738)

(48.653532815138554, -0.000422802805149809)

(-95.7976925376814, -0.000108989467051541)

(-36.072758289477676, 0.0007696756958536)

(10.806227045602704, -0.0087087677703681)

(32.92583070183405, 0.000924115453595151)

(-32.92583070183405, -0.000924115453595151)

(98.9399487990654, -0.000102175165394175)

(-17.16115782561268, 0.00341850130927523)

(202.62285496058013, 2.43581496647271e-5)

(36.072758289477676, -0.0007696756958536)

(92.65539121488297, -0.000116509077378612)

(-73.80031393214543, 0.000183672018953665)

(-29.777763739304003, 0.00113029856328114)

(-70.65751403023398, -0.000200381305938913)

(-155.49597286464936, 4.1361628405071e-5)

(26.62814611127368, 0.00141429382702696)

(76.94301503968819, 0.000168969479119145)

(-76.94301503968819, -0.000168969479119145)

(-51.79762858398812, -0.000372995628973321)

(20.32129595212744, 0.00243326959458963)

(-61.22836890560009, 0.000266886242151348)

(86.37063367626389, -0.000134086233502956)

(-54.94143686927333, 0.000331503052571738)

(-92.65539121488297, 0.000116509077378612)

(51.79762858398812, 0.000372995628973321)

(13.992272290899354, 0.00515956885743496)

(-89.51304007684683, -0.000124834826472013)

(39.21882376823015, 0.000650990789484618)

(29.777763739304003, -0.00113029856328114)


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=23.4763396531088x_{1} = 23.4763396531088
x2=4.15222477298826x_{2} = 4.15222477298826
x3=80.0856290117635x_{3} = 80.0856290117635
x4=−58.0850045720611x_{4} = -58.0850045720611
x5=73.8003139321454x_{5} = 73.8003139321454
x6=−26.6281461112737x_{6} = -26.6281461112737
x7=−13.9922722908994x_{7} = -13.9922722908994
x8=−83.2281657410082x_{8} = -83.2281657410082
x9=67.51460149576x_{9} = 67.51460149576
x10=17.1611578256127x_{10} = 17.1611578256127
x11=−64.3715597764874x_{11} = -64.3715597764874
x12=−45.5090895943713x_{12} = -45.5090895943713
x13=61.2283689056001x_{13} = 61.2283689056001
x14=−39.2188237682302x_{14} = -39.2188237682302
x15=−290.590437700776x_{15} = -290.590437700776
x16=42.3642209820001x_{16} = 42.3642209820001
x17=−7.57754404149509x_{17} = -7.57754404149509
x18=−20.3212959521274x_{18} = -20.3212959521274
x19=54.9414368692733x_{19} = 54.9414368692733
x20=48.6535328151386x_{20} = 48.6535328151386
x21=−95.7976925376814x_{21} = -95.7976925376814
x22=10.8062270456027x_{22} = 10.8062270456027
x23=−32.9258307018341x_{23} = -32.9258307018341
x24=98.9399487990654x_{24} = 98.9399487990654
x25=36.0727582894777x_{25} = 36.0727582894777
x26=92.655391214883x_{26} = 92.655391214883
x27=−70.657514030234x_{27} = -70.657514030234
x28=−76.9430150396882x_{28} = -76.9430150396882
x29=−51.7976285839881x_{29} = -51.7976285839881
x30=86.3706336762639x_{30} = 86.3706336762639
x31=−89.5130400768468x_{31} = -89.5130400768468
x32=29.777763739304x_{32} = 29.777763739304
Puntos máximos de la función:
x32=45.5090895943713x_{32} = 45.5090895943713
x32=−80.0856290117635x_{32} = -80.0856290117635
x32=64.3715597764874x_{32} = 64.3715597764874
x32=70.657514030234x_{32} = 70.657514030234
x32=−23.4763396531088x_{32} = -23.4763396531088
x32=−10.8062270456027x_{32} = -10.8062270456027
x32=−42.3642209820001x_{32} = -42.3642209820001
x32=89.5130400768468x_{32} = 89.5130400768468
x32=−98.9399487990654x_{32} = -98.9399487990654
x32=−67.51460149576x_{32} = -67.51460149576
x32=−48.6535328151386x_{32} = -48.6535328151386
x32=−4.15222477298826x_{32} = -4.15222477298826
x32=95.7976925376814x_{32} = 95.7976925376814
x32=7.57754404149509x_{32} = 7.57754404149509
x32=−86.3706336762639x_{32} = -86.3706336762639
x32=83.2281657410082x_{32} = 83.2281657410082
x32=58.0850045720611x_{32} = 58.0850045720611
x32=−124.076787974095x_{32} = -124.076787974095
x32=−36.0727582894777x_{32} = -36.0727582894777
x32=32.9258307018341x_{32} = 32.9258307018341
x32=−17.1611578256127x_{32} = -17.1611578256127
x32=202.62285496058x_{32} = 202.62285496058
x32=−73.8003139321454x_{32} = -73.8003139321454
x32=−29.777763739304x_{32} = -29.777763739304
x32=−155.495972864649x_{32} = -155.495972864649
x32=26.6281461112737x_{32} = 26.6281461112737
x32=76.9430150396882x_{32} = 76.9430150396882
x32=20.3212959521274x_{32} = 20.3212959521274
x32=−61.2283689056001x_{32} = -61.2283689056001
x32=−54.9414368692733x_{32} = -54.9414368692733
x32=−92.655391214883x_{32} = -92.655391214883
x32=51.7976285839881x_{32} = 51.7976285839881
x32=13.9922722908994x_{32} = 13.9922722908994
x32=39.2188237682302x_{32} = 39.2188237682302
Decrece en los intervalos
[98.9399487990654,∞)\left[98.9399487990654, \infty\right)
Crece en los intervalos
(−∞,−290.590437700776]\left(-\infty, -290.590437700776\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)x2−4−sin⁡(x)+2(4x2x2−4−1)sin⁡(x)x2−4x2−4=0\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−34.4409194318359x_{1} = -34.4409194318359
x2=−69.0570585223772x_{2} = -69.0570585223772
x3=−81.6323744858714x_{3} = -81.6323744858714
x4=5.40521123343275x_{4} = 5.40521123343275
x5=−103.633943397032x_{5} = -103.633943397032
x6=91.0622362167619x_{6} = 91.0622362167619
x7=−72.2011806345949x_{7} = -72.2011806345949
x8=65.9126941265009x_{8} = 65.9126941265009
x9=−65.9126941265009x_{9} = -65.9126941265009
x10=21.80589274806x_{10} = 21.80589274806
x11=−100.491142008147x_{11} = -100.491142008147
x12=−12.2287224949723x_{12} = -12.2287224949723
x13=119.346994135494x_{13} = 119.346994135494
x14=−37.5923546185444x_{14} = -37.5923546185444
x15=−59.6230841163499x_{15} = -59.6230841163499
x16=−87.91907042905x_{16} = -87.91907042905
x17=62.768050805063x_{17} = 62.768050805063
x18=28.13129974445x_{18} = 28.13129974445
x19=87.91907042905x_{19} = 87.91907042905
x20=97.348262428604x_{20} = 97.348262428604
x21=24.9713492935255x_{21} = 24.9713492935255
x22=−31.2874672441028x_{22} = -31.2874672441028
x23=31.2874672441028x_{23} = 31.2874672441028
x24=56.4777396209399x_{24} = 56.4777396209399
x25=84.7757877051701x_{25} = 84.7757877051701
x26=−28.13129974445x_{26} = -28.13129974445
x27=−56.4777396209399x_{27} = -56.4777396209399
x28=−47.0386736392297x_{28} = -47.0386736392297
x29=15.4437856317982x_{29} = 15.4437856317982
x30=−43.890940829429x_{30} = -43.890940829429
x31=50.1856303934487x_{31} = 50.1856303934487
x32=−84.7757877051701x_{32} = -84.7757877051701
x33=−97.348262428604x_{33} = -97.348262428604
x34=72.2011806345949x_{34} = 72.2011806345949
x35=−109.919338493527x_{35} = -109.919338493527
x36=−5.40521123343275x_{36} = -5.40521123343275
x37=43.890940829429x_{37} = 43.890940829429
x38=−53.3319498938786x_{38} = -53.3319498938786
x39=−40.7422494725529x_{39} = -40.7422494725529
x40=37.5923546185444x_{40} = 37.5923546185444
x41=0x_{41} = 0
x42=−15.4437856317982x_{42} = -15.4437856317982
x43=94.2052968101638x_{43} = 94.2052968101638
x44=−75.3450909469212x_{44} = -75.3450909469212
x45=−62.768050805063x_{45} = -62.768050805063
x46=−91.0622362167619x_{46} = -91.0622362167619
x47=−24.9713492935255x_{47} = -24.9713492935255
x48=78.4888150282294x_{48} = 78.4888150282294
x49=−78.4888150282294x_{49} = -78.4888150282294
x50=−21.80589274806x_{50} = -21.80589274806
x51=81.6323744858714x_{51} = 81.6323744858714
x52=18.6319434077619x_{52} = 18.6319434077619
x53=−8.95019260991225x_{53} = -8.95019260991225
x54=75.3450909469212x_{54} = 75.3450909469212
x55=69.0570585223772x_{55} = 69.0570585223772
x56=−50.1856303934487x_{56} = -50.1856303934487
x57=34.4409194318359x_{57} = 34.4409194318359
x58=59.6230841163499x_{58} = 59.6230841163499
x59=47.0386736392297x_{59} = 47.0386736392297
x60=−94.2052968101638x_{60} = -94.2052968101638
x61=40.7422494725529x_{61} = 40.7422494725529
x62=53.3319498938786x_{62} = 53.3319498938786
x63=100.491142008147x_{63} = 100.491142008147
x64=166.480379738649x_{64} = 166.480379738649
x65=−18.6319434077619x_{65} = -18.6319434077619
x66=12.2287224949723x_{66} = 12.2287224949723
x67=8.95019260991225x_{67} = 8.95019260991225
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=−2x_{1} = -2
x2=2x_{2} = 2

lim⁡x→−2−(−4xcos⁡(x)x2−4−sin⁡(x)+2(4x2x2−4−1)sin⁡(x)x2−4x2−4)=−∞\lim_{x \to -2^-}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = -\infty
lim⁡x→−2+(−4xcos⁡(x)x2−4−sin⁡(x)+2(4x2x2−4−1)sin⁡(x)x2−4x2−4)=∞\lim_{x \to -2^+}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = \infty
- los límites no son iguales, signo
x1=−2x_{1} = -2
- es el punto de flexión
lim⁡x→2−(−4xcos⁡(x)x2−4−sin⁡(x)+2(4x2x2−4−1)sin⁡(x)x2−4x2−4)=−∞\lim_{x \to 2^-}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = -\infty
lim⁡x→2+(−4xcos⁡(x)x2−4−sin⁡(x)+2(4x2x2−4−1)sin⁡(x)x2−4x2−4)=∞\lim_{x \to 2^+}\left(\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 4} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 4} - 1\right) \sin{\left(x \right)}}{x^{2} - 4}}{x^{2} - 4}\right) = \infty
- los límites no son iguales, signo
x2=2x_{2} = 2
- 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
[166.480379738649,∞)\left[166.480379738649, \infty\right)
Convexa en los intervalos
(−∞,−109.919338493527]\left(-\infty, -109.919338493527\right]
Asíntotas verticales
Hay:
x1=−2x_{1} = -2
x2=2x_{2} = 2
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(x)x2−4)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x^{2} - 4}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(x)x2−4)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x^{2} - 4}\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)/(x^2 - 4), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x)x(x2−4))=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(x^{2} - 4\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(x)x(x2−4))=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(x^{2} - 4\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)x2−4=−sin⁡(x)x2−4\frac{\sin{\left(x \right)}}{x^{2} - 4} = - \frac{\sin{\left(x \right)}}{x^{2} - 4}
- No
sin⁡(x)x2−4=sin⁡(x)x2−4\frac{\sin{\left(x \right)}}{x^{2} - 4} = \frac{\sin{\left(x \right)}}{x^{2} - 4}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор