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

Gráfico de la función y = sinx\(x(x-2))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         sin(x) 
f(x) = ---------
       x*(x - 2)
f(x)=sin⁡(x)x(x−2)f{\left(x \right)} = \frac{\sin{\left(x \right)}}{x \left(x - 2\right)} 
f = sin(x)/((x*(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=0x_{1} = 0
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)x(x−2)=0\frac{\sin{\left(x \right)}}{x \left(x - 2\right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=πx_{1} = \pi
Solución numérica
x1=−138.230076757951x_{1} = -138.230076757951
x2=40.8407044966673x_{2} = 40.8407044966673
x3=−18.8495559215388x_{3} = -18.8495559215388
x4=−56.5486677646163x_{4} = -56.5486677646163
x5=97.3893722612836x_{5} = 97.3893722612836
x6=34.5575191894877x_{6} = 34.5575191894877
x7=53.4070751110265x_{7} = 53.4070751110265
x8=47.1238898038469x_{8} = 47.1238898038469
x9=−97.3893722612836x_{9} = -97.3893722612836
x10=62.8318530717959x_{10} = 62.8318530717959
x11=87.9645943005142x_{11} = 87.9645943005142
x12=43.9822971502571x_{12} = 43.9822971502571
x13=37.6991118430775x_{13} = 37.6991118430775
x14=−21.9911485751286x_{14} = -21.9911485751286
x15=65.9734457253857x_{15} = 65.9734457253857
x16=69.1150383789755x_{16} = 69.1150383789755
x17=−50.2654824574367x_{17} = -50.2654824574367
x18=−94.2477796076938x_{18} = -94.2477796076938
x19=−75.398223686155x_{19} = -75.398223686155
x20=−53.4070751110265x_{20} = -53.4070751110265
x21=12.5663706143592x_{21} = 12.5663706143592
x22=−9.42477796076938x_{22} = -9.42477796076938
x23=−8033.0524152291x_{23} = -8033.0524152291
x24=−703.716754404114x_{24} = -703.716754404114
x25=−34.5575191894877x_{25} = -34.5575191894877
x26=21.9911485751286x_{26} = 21.9911485751286
x27=−116.238928182822x_{27} = -116.238928182822
x28=−47.1238898038469x_{28} = -47.1238898038469
x29=−43.9822971502571x_{29} = -43.9822971502571
x30=28.2743338823081x_{30} = 28.2743338823081
x31=−31.4159265358979x_{31} = -31.4159265358979
x32=−3.14159265358979x_{32} = -3.14159265358979
x33=−6.28318530717959x_{33} = -6.28318530717959
x34=−25.1327412287183x_{34} = -25.1327412287183
x35=−62.8318530717959x_{35} = -62.8318530717959
x36=31.4159265358979x_{36} = 31.4159265358979
x37=−65.9734457253857x_{37} = -65.9734457253857
x38=72.2566310325652x_{38} = 72.2566310325652
x39=−59.6902604182061x_{39} = -59.6902604182061
x40=94.2477796076938x_{40} = 94.2477796076938
x41=81.6814089933346x_{41} = 81.6814089933346
x42=−91.106186954104x_{42} = -91.106186954104
x43=−100.530964914873x_{43} = -100.530964914873
x44=59.6902604182061x_{44} = 59.6902604182061
x45=−40.8407044966673x_{45} = -40.8407044966673
x46=91.106186954104x_{46} = 91.106186954104
x47=78.5398163397448x_{47} = 78.5398163397448
x48=−12.5663706143592x_{48} = -12.5663706143592
x49=56.5486677646163x_{49} = 56.5486677646163
x50=84.8230016469244x_{50} = 84.8230016469244
x51=100.530964914873x_{51} = 100.530964914873
x52=−69.1150383789755x_{52} = -69.1150383789755
x53=9.42477796076938x_{53} = 9.42477796076938
x54=116.238928182822x_{54} = 116.238928182822
x55=−84.8230016469244x_{55} = -84.8230016469244
x56=−78.5398163397448x_{56} = -78.5398163397448
x57=−87.9645943005142x_{57} = -87.9645943005142
x58=−81.6814089933346x_{58} = -81.6814089933346
x59=15.707963267949x_{59} = 15.707963267949
x60=−28.2743338823081x_{60} = -28.2743338823081
x61=−15.707963267949x_{61} = -15.707963267949
x62=−37.6991118430775x_{62} = -37.6991118430775
x63=18.8495559215388x_{63} = 18.8495559215388
x64=25.1327412287183x_{64} = 25.1327412287183
x65=50.2654824574367x_{65} = 50.2654824574367
x66=−72.2566310325652x_{66} = -72.2566310325652
x67=75.398223686155x_{67} = 75.398223686155
x68=6.28318530717959x_{68} = 6.28318530717959
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*(x - 2))).
sin⁡(0)(−2)0\frac{\sin{\left(0 \right)}}{\left(-2\right) 0}
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
1x(x−2)cos⁡(x)+(2−2x)sin⁡(x)x2(x−2)2=0\frac{1}{x \left(x - 2\right)} \cos{\left(x \right)} + \frac{\left(2 - 2 x\right) \sin{\left(x \right)}}{x^{2} \left(x - 2\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=32.9240947361922x_{1} = 32.9240947361922
x2=−51.7984033793852x_{2} = -51.7984033793852
x3=17.1551175692195x_{3} = 17.1551175692195
x4=36.0713042845073x_{4} = 36.0713042845073
x5=−98.9401572801928x_{5} = -98.9401572801928
x6=−20.3266414701876x_{6} = -20.3266414701876
x7=39.2175881820314x_{7} = 39.2175881820314
x8=−42.3653892667121x_{8} = -42.3653892667121
x9=−80.0859487309835x_{9} = -80.0859487309835
x10=45.5081654611057x_{10} = 45.5081654611057
x11=−10.8268690995624x_{11} = -10.8268690995624
x12=80.0853248723334x_{12} = 80.0853248723334
x13=−70.6579261366639x_{13} = -70.6579261366639
x14=67.5141755390553x_{14} = 67.5141755390553
x15=−7.62307729555873x_{15} = -7.62307729555873
x16=95.7974791095636x_{16} = 95.7974791095636
x17=−17.1687916817231x_{17} = -17.1687916817231
x18=−58.0856181381215x_{18} = -58.0856181381215
x19=−186.914119829292x_{19} = -186.914119829292
x20=−92.6556292628207x_{20} = -92.6556292628207
x21=10.7920322357124x_{21} = 10.7920322357124
x22=190.055776752248x_{22} = 190.055776752248
x23=−76.943361762789x_{23} = -76.943361762789
x24=4.08557388547682x_{24} = 4.08557388547682
x25=−86.3709080585407x_{25} = -86.3709080585407
x26=83.2278838695937x_{26} = 83.2278838695937
x27=64.3710918861937x_{27} = 64.3710918861937
x28=−32.927791200115x_{28} = -32.927791200115
x29=−256.032020175128x_{29} = -256.032020175128
x30=−36.0743829885085x_{30} = -36.0743829885085
x31=−61.2289201135729x_{31} = -61.2289201135729
x32=−83.2284614928992x_{32} = -83.2284614928992
x33=−39.220192145926x_{33} = -39.220192145926
x34=−73.8006912312722x_{34} = -73.8006912312722
x35=−23.4802919264971x_{35} = -23.4802919264971
x36=51.7969113967309x_{36} = 51.7969113967309
x37=29.7756549714707x_{37} = 29.7756549714707
x38=7.55102453615362x_{38} = 7.55102453615362
x39=48.6527219697238x_{39} = 48.6527219697238
x40=58.0844318525335x_{40} = 58.0844318525335
x41=−26.6311871536774x_{41} = -26.6311871536774
x42=−64.3720576734236x_{42} = -64.3720576734236
x43=76.9426858848596x_{43} = 76.9426858848596
x44=61.2278525685536x_{44} = 61.2278525685536
x45=23.4730079186956x_{45} = 23.4730079186956
x46=−89.5132953248997x_{46} = -89.5132953248997
x47=73.7999565431799x_{47} = 73.7999565431799
x48=92.6551632265069x_{48} = 92.6551632265069
x49=26.6255299708041x_{49} = 26.6255299708041
x50=−95.7979150674353x_{50} = -95.7979150674353
x51=−29.7801761695834x_{51} = -29.7801761695834
x52=−48.6544131811461x_{52} = -48.6544131811461
x53=13.9834279458844x_{53} = 13.9834279458844
x54=89.5127959855887x_{54} = 89.5127959855887
x55=98.9397485795284x_{55} = 98.9397485795284
x56=86.3703717136003x_{56} = 86.3703717136003
x57=−14.0040665914265x_{57} = -14.0040665914265
x58=70.6571246114187x_{58} = 70.6571246114187
x59=20.3169083532025x_{59} = 20.3169083532025
x60=−45.5100986876418x_{60} = -45.5100986876418
x61=1371.30373376177x_{61} = 1371.30373376177
x62=−488.513571950338x_{62} = -488.513571950338
x63=54.9407979981311x_{63} = 54.9407979981311
x64=−67.5150534592896x_{64} = -67.5150534592896
x65=42.3631580330254x_{65} = 42.3631580330254
x66=−54.9421240104386x_{66} = -54.9421240104386
x67=−4.34230123285199x_{67} = -4.34230123285199
Signos de extremos en los puntos:
(32.92409473619218, 0.000980250092940406)

(-51.79840337938525, -0.000358593701721455)

(17.155117569219506, -0.00381697128504473)

(36.07130428450728, -0.000812350070903317)

(-98.94015728019278, 0.00010010976999049)

(-20.326641470187568, -0.00219382132146474)

(39.21758818203142, 0.000684189186675888)

(-42.365389266712086, 0.000531474734651108)

(-80.08594873098346, 0.000152069720255664)

(45.50816546110567, 0.000504546775560921)

(-10.826869099562385, 0.00709850000065266)

(80.08532487233344, -0.000159859614862926)

(-70.6579261366639, -0.000194709418827277)

(67.51417553905529, -0.000225981778238015)

(-7.623077295558733, -0.0132700859723812)

(95.79747910956362, 0.000111264878610497)

(-17.16879168172315, 0.00302019005045724)

(-58.08561813812148, -0.000286359800368288)

(-186.91411982929165, 2.83184105065328e-5)

(-92.65562926282065, 0.000113994190199678)

(10.792032235712393, -0.0103216328869677)

(190.055776752248, 2.7977438203328e-5)

(-76.94336176278898, -0.000164577420962203)

(4.085573885476822, -0.0950501114395919)

(-86.37090805854075, 0.000130981335117623)

(83.22788386959367, 0.00014787594936823)

(64.37109188619371, 0.000248948485979724)

(-32.92779120011495, -0.000867983965173453)

(-256.0320201751279, 1.51362735214327e-5)

(-36.074382988508454, 0.00072700314182954)

(-61.22892011357289, 0.000258168549826435)

(-83.22846149289921, -0.000140935711429685)

(-39.220192145926006, -0.000617793494390805)

(-73.80069123127224, 0.000178694491361609)

(-23.480291926497095, 0.00166587622836872)

(51.79691139673087, 0.000387397763643712)

(29.775654971470725, -0.00120621686240201)

(7.551024536153622, 0.0227707843461169)

(48.652721969723785, -0.000440183098167783)

(58.08443185253347, 0.00030678306311284)

(-26.631187153677416, -0.00130807441032797)

(-64.37205767342357, -0.000233945166452643)

(76.94268588485963, 0.000173361556566348)

(61.227852568553566, -0.000275604010475087)

(23.473007918695554, -0.0019761436581552)

(-89.51329532489973, -0.000122045631771871)

(73.79995654317989, -0.000188649571321626)

(92.65516322650693, -0.000119023970882221)

(26.625529970804127, 0.00152052452751166)

(-95.79791506743528, -0.000106714060670071)

(-29.78017616958336, 0.00105438602731828)

(-48.654413181146104, 0.000405422814201224)

(13.983427945884438, 0.00589729038830167)

(89.51279598558865, 0.000127624028951783)

(98.93974857952836, -0.000104240565063727)

(86.3703717136003, -0.000137191141528712)

(-14.004066591426517, -0.00442239117312082)

(70.65712461141868, 0.000206053225221036)

(20.316908353202518, 0.00267277521726431)

(-45.51009868764183, -0.000462066792988833)

(1371.3037337617723, 5.32556986595609e-7)

(-488.51357195033813, 4.17319533301501e-6)

(54.940797998131075, -0.00034357062848065)

(-67.5150534592896, 0.000212978367092361)

(42.363158033025364, -0.000584142570509146)

(-54.942124010438576, 0.000319435622287472)

(-4.3423012328519945, 0.033852194422576)


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=−51.7984033793852x_{1} = -51.7984033793852
x2=17.1551175692195x_{2} = 17.1551175692195
x3=36.0713042845073x_{3} = 36.0713042845073
x4=−20.3266414701876x_{4} = -20.3266414701876
x5=80.0853248723334x_{5} = 80.0853248723334
x6=−70.6579261366639x_{6} = -70.6579261366639
x7=67.5141755390553x_{7} = 67.5141755390553
x8=−7.62307729555873x_{8} = -7.62307729555873
x9=−58.0856181381215x_{9} = -58.0856181381215
x10=10.7920322357124x_{10} = 10.7920322357124
x11=−76.943361762789x_{11} = -76.943361762789
x12=4.08557388547682x_{12} = 4.08557388547682
x13=−32.927791200115x_{13} = -32.927791200115
x14=−83.2284614928992x_{14} = -83.2284614928992
x15=−39.220192145926x_{15} = -39.220192145926
x16=29.7756549714707x_{16} = 29.7756549714707
x17=48.6527219697238x_{17} = 48.6527219697238
x18=−26.6311871536774x_{18} = -26.6311871536774
x19=−64.3720576734236x_{19} = -64.3720576734236
x20=61.2278525685536x_{20} = 61.2278525685536
x21=23.4730079186956x_{21} = 23.4730079186956
x22=−89.5132953248997x_{22} = -89.5132953248997
x23=73.7999565431799x_{23} = 73.7999565431799
x24=92.6551632265069x_{24} = 92.6551632265069
x25=−95.7979150674353x_{25} = -95.7979150674353
x26=98.9397485795284x_{26} = 98.9397485795284
x27=86.3703717136003x_{27} = 86.3703717136003
x28=−14.0040665914265x_{28} = -14.0040665914265
x29=−45.5100986876418x_{29} = -45.5100986876418
x30=54.9407979981311x_{30} = 54.9407979981311
x31=42.3631580330254x_{31} = 42.3631580330254
Puntos máximos de la función:
x31=32.9240947361922x_{31} = 32.9240947361922
x31=−98.9401572801928x_{31} = -98.9401572801928
x31=39.2175881820314x_{31} = 39.2175881820314
x31=−42.3653892667121x_{31} = -42.3653892667121
x31=−80.0859487309835x_{31} = -80.0859487309835
x31=45.5081654611057x_{31} = 45.5081654611057
x31=−10.8268690995624x_{31} = -10.8268690995624
x31=95.7974791095636x_{31} = 95.7974791095636
x31=−17.1687916817231x_{31} = -17.1687916817231
x31=−186.914119829292x_{31} = -186.914119829292
x31=−92.6556292628207x_{31} = -92.6556292628207
x31=190.055776752248x_{31} = 190.055776752248
x31=−86.3709080585407x_{31} = -86.3709080585407
x31=83.2278838695937x_{31} = 83.2278838695937
x31=64.3710918861937x_{31} = 64.3710918861937
x31=−256.032020175128x_{31} = -256.032020175128
x31=−36.0743829885085x_{31} = -36.0743829885085
x31=−61.2289201135729x_{31} = -61.2289201135729
x31=−73.8006912312722x_{31} = -73.8006912312722
x31=−23.4802919264971x_{31} = -23.4802919264971
x31=51.7969113967309x_{31} = 51.7969113967309
x31=7.55102453615362x_{31} = 7.55102453615362
x31=58.0844318525335x_{31} = 58.0844318525335
x31=76.9426858848596x_{31} = 76.9426858848596
x31=26.6255299708041x_{31} = 26.6255299708041
x31=−29.7801761695834x_{31} = -29.7801761695834
x31=−48.6544131811461x_{31} = -48.6544131811461
x31=13.9834279458844x_{31} = 13.9834279458844
x31=89.5127959855887x_{31} = 89.5127959855887
x31=70.6571246114187x_{31} = 70.6571246114187
x31=20.3169083532025x_{31} = 20.3169083532025
x31=1371.30373376177x_{31} = 1371.30373376177
x31=−488.513571950338x_{31} = -488.513571950338
x31=−67.5150534592896x_{31} = -67.5150534592896
x31=−54.9421240104386x_{31} = -54.9421240104386
x31=−4.34230123285199x_{31} = -4.34230123285199
Decrece en los intervalos
[98.9397485795284,∞)\left[98.9397485795284, \infty\right)
Crece en los intervalos
(−∞,−95.7979150674353]\left(-\infty, -95.7979150674353\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
−sin⁡(x)−4(x−1)cos⁡(x)x(x−2)+2((x−1)(1x−2+1x)−1+x−1x−2+x−1x)sin⁡(x)x(x−2)x(x−2)=0\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=94.2048551569328x_{1} = 94.2048551569328
x2=37.5896554681014x_{2} = 37.5896554681014
x3=−94.2057576185378x_{3} = -94.2057576185378
x4=−18.6450684444873x_{4} = -18.6450684444873
x5=21.79808611927x_{5} = 21.79808611927
x6=−65.9136449449715x_{6} = -65.9136449449715
x7=91.0617638658387x_{7} = 91.0617638658387
x8=8.90587840675485x_{8} = 8.90587840675485
x9=−28.1367815741316x_{9} = -28.1367815741316
x10=50.1840993820307x_{10} = 50.1840993820307
x11=−97.3486936418635x_{11} = -97.3486936418635
x12=15.4285406261646x_{12} = 15.4285406261646
x13=81.6317880542076x_{13} = 81.6317880542076
x14=59.6219935715278x_{14} = 59.6219935715278
x15=47.0369347933021x_{15} = 47.0369347933021
x16=−84.7763581906396x_{16} = -84.7763581906396
x17=−59.6242503189044x_{17} = -59.6242503189044
x18=−109.919675859272x_{18} = -109.919675859272
x19=−21.815268529783x_{19} = -21.815268529783
x20=−9.01907137188613x_{20} = -9.01907137188613
x21=87.9185640627454x_{21} = 87.9185640627454
x22=65.9117992853432x_{22} = 65.9117992853432
x23=−50.1872883920623x_{23} = -50.1872883920623
x24=75.3444038224365x_{24} = 75.3444038224365
x25=62.7670653675399x_{25} = 62.7670653675399
x26=−47.0405667699869x_{26} = -47.0405667699869
x27=97.347848573433x_{27} = 97.347848573433
x28=−31.2918587569744x_{28} = -31.2918587569744
x29=−12.2619358385377x_{29} = -12.2619358385377
x30=84.7752435095284x_{30} = 84.7752435095284
x31=−87.9196003632094x_{31} = -87.9196003632094
x32=−37.5953566524211x_{32} = -37.5953566524211
x33=−53.3334140575947x_{33} = -53.3334140575947
x34=−116.204795746496x_{34} = -116.204795746496
x35=−40.7447926388754x_{35} = -40.7447926388754
x36=298.437853650273x_{36} = 298.437853650273
x37=31.2836024135663x_{37} = 31.2836024135663
x38=40.7399440312312x_{38} = 40.7399440312312
x39=100.490753405596x_{39} = 100.490753405596
x40=−122.489718589867x_{40} = -122.489718589867
x41=−91.0627297768666x_{41} = -91.0627297768666
x42=−78.4894819481931x_{42} = -78.4894819481931
x43=−141.343566076059x_{43} = -141.343566076059
x44=24.9653511953722x_{44} = 24.9653511953722
x45=43.8889486335222x_{45} = 43.8889486335222
x46=−75.345815531326x_{46} = -75.345815531326
x47=18.6213492879788x_{47} = 18.6213492879788
x48=56.4765261792329x_{48} = 56.4765261792329
x49=−69.0579233879794x_{49} = -69.0579233879794
x50=5.2766637040869x_{50} = 5.2766637040869
x51=78.4881812412389x_{51} = 78.4881812412389
x52=−34.4445173164217x_{52} = -34.4445173164217
x53=28.126543898197x_{53} = 28.126543898197
x54=−100.491546389739x_{54} = -100.491546389739
x55=−62.7691010739784x_{55} = -62.7691010739784
x56=−376.980535751818x_{56} = -376.980535751818
x57=−5.66149669154895x_{57} = -5.66149669154895
x58=−24.9783881327673x_{58} = -24.9783881327673
x59=−56.4790421008596x_{59} = -56.4790421008596
x60=−81.6329903654042x_{60} = -81.6329903654042
x61=69.056242322581x_{61} = 69.056242322581
x62=34.4377158397736x_{62} = 34.4377158397736
x63=−72.2019707064293x_{63} = -72.2019707064293
x64=12.2047300409779x_{64} = 12.2047300409779
x65=−43.8931230440976x_{65} = -43.8931230440976
x66=−103.63432337662x_{66} = -103.63432337662
x67=53.3305915039441x_{67} = 53.3305915039441
x68=72.2004331376217x_{68} = 72.2004331376217
x69=−15.4635207898039x_{69} = -15.4635207898039
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
x2=2x_{2} = 2

lim⁡x→0−(−sin⁡(x)−4(x−1)cos⁡(x)x(x−2)+2((x−1)(1x−2+1x)−1+x−1x−2+x−1x)sin⁡(x)x(x−2)x(x−2))=−112\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\right) = - \frac{1}{12}
lim⁡x→0+(−sin⁡(x)−4(x−1)cos⁡(x)x(x−2)+2((x−1)(1x−2+1x)−1+x−1x−2+x−1x)sin⁡(x)x(x−2)x(x−2))=−112\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\right) = - \frac{1}{12}
- los límites son iguales, es decir omitimos el punto correspondiente
lim⁡x→2−(−sin⁡(x)−4(x−1)cos⁡(x)x(x−2)+2((x−1)(1x−2+1x)−1+x−1x−2+x−1x)sin⁡(x)x(x−2)x(x−2))=−∞\lim_{x \to 2^-}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\right) = -\infty
lim⁡x→2+(−sin⁡(x)−4(x−1)cos⁡(x)x(x−2)+2((x−1)(1x−2+1x)−1+x−1x−2+x−1x)sin⁡(x)x(x−2)x(x−2))=∞\lim_{x \to 2^+}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\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
[298.437853650273,∞)\left[298.437853650273, \infty\right)
Convexa en los intervalos
(−∞,−141.343566076059]\left(-\infty, -141.343566076059\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
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)x(x−2))=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(x - 2\right)}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(x)x(x−2))=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(x - 2\right)}\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*(x - 2))), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(1x(x−2)sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{x \left(x - 2\right)} \sin{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(1x(x−2)sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{x \left(x - 2\right)} \sin{\left(x \right)}}{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:
sin⁡(x)x(x−2)=sin⁡(x)x(−x−2)\frac{\sin{\left(x \right)}}{x \left(x - 2\right)} = \frac{\sin{\left(x \right)}}{x \left(- x - 2\right)}
- No
sin⁡(x)x(x−2)=−sin⁡(x)x(−x−2)\frac{\sin{\left(x \right)}}{x \left(x - 2\right)} = - \frac{\sin{\left(x \right)}}{x \left(- x - 2\right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор