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acot(x)/(1+x^2)
Trazado el gráfico de la función/ acot(x)/(1+x^2)

Gráfico de la función y = acot(x)/(1+x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       acot(x)
f(x) = -------
             2
        1 + x
f(x)=acot⁡(x)x2+1f{\left(x \right)} = \frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 1} 
f = acot(x)/(x^2 + 1)
Gráfico de la función
02468-8-6-4-2-10105-5
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:
acot⁡(x)x2+1=0\frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 1} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−27061.149062433x_{1} = -27061.149062433
x2=37362.6871895938x_{2} = 37362.6871895938
x3=11092.8422511079x_{3} = 11092.8422511079
x4=39905.3415954009x_{4} = 39905.3415954009
x5=39057.7878366038x_{5} = 39057.7878366038
x6=14481.2994735822x_{6} = 14481.2994735822
x7=−10961.7090219324x_{7} = -10961.7090219324
x8=36515.1406203303x_{8} = 36515.1406203303
x9=−29603.6671490675x_{9} = -29603.6671490675
x10=−25366.1698184569x_{10} = -25366.1698184569
x11=−39774.116186012x_{11} = -39774.116186012
x12=−31298.7058008722x_{12} = -31298.7058008722
x13=−30451.1841116306x_{13} = -30451.1841116306
x14=−21976.317416806x_{14} = -21976.317416806
x15=33124.9838542871x_{15} = 33124.9838542871
x16=−26213.6557347134x_{16} = -26213.6557347134
x17=28039.8666749084x_{17} = 28039.8666749084
x18=17023.1545521275x_{18} = 17023.1545521275
x19=30582.4041363405x_{19} = 30582.4041363405
x20=32277.453202895x_{20} = 32277.453202895
x21=−37231.4628580868x_{21} = -37231.4628580868
x22=18717.8691313136x_{22} = 18717.8691313136
x23=17870.5001302529x_{23} = 17870.5001302529
x24=12786.9409888948x_{24} = 12786.9409888948
x25=21260.0886635237x_{25} = 21260.0886635237
x26=26344.8712183295x_{26} = 26344.8712183295
x27=−13502.9266850983x_{27} = -13502.9266850983
x28=−28756.1553311053x_{28} = -28756.1553311053
x29=31429.9265202585x_{29} = 31429.9265202585
x30=−35536.3733106874x_{30} = -35536.3733106874
x31=41600.4552836757x_{31} = 41600.4552836757
x32=−16891.9636999165x_{32} = -16891.9636999165
x33=34820.0559041106x_{33} = 34820.0559041106
x34=−34688.8328956213x_{34} = -34688.8328956213
x35=23802.4349222602x_{35} = 23802.4349222602
x36=−18586.6709575375x_{36} = -18586.6709575375
x37=11939.8522404524x_{37} = 11939.8522404524
x38=−27908.6491264287x_{38} = -27908.6491264287
x39=−11808.7052724478x_{39} = -11808.7052724478
x40=19565.2585118578x_{40} = 19565.2585118578
x41=−38926.5627631727x_{41} = -38926.5627631727
x42=33972.5181773981x_{42} = 33972.5181773981
x43=−24518.692082196x_{43} = -24518.692082196
x44=13634.0938154046x_{44} = 13634.0938154046
x45=−32993.7619023979x_{45} = -32993.7619023979
x46=22107.5254947614x_{46} = 22107.5254947614
x47=−10114.8146354757x_{47} = -10114.8146354757
x48=−12655.7829341849x_{48} = -12655.7829341849
x49=−38079.0116023705x_{49} = -38079.0116023705
x50=20412.6657337582x_{50} = 20412.6657337582
x51=27192.3656268491x_{51} = 27192.3656268491
x52=35667.5967917905x_{52} = 35667.5967917905
x53=−21128.8826230279x_{53} = -21128.8826230279
x54=25497.38411143x_{54} = 25497.38411143
x55=−16044.6497827596x_{55} = -16044.6497827596
x56=−36383.9166991972x_{56} = -36383.9166991972
x57=−40621.6717292993x_{57} = -40621.6717292993
x58=42448.0149667315x_{58} = 42448.0149667315
x59=16175.8360780369x_{59} = 16175.8360780369
x60=10245.9305554705x_{60} = 10245.9305554705
x61=−14350.1248203195x_{61} = -14350.1248203195
x62=28887.3737781207x_{62} = 28887.3737781207
x63=−9268.05003936139x_{63} = -9268.05003936139
x64=−15197.3682444233x_{64} = -15197.3682444233
x65=−33841.2956774047x_{65} = -33841.2956774047
x66=40752.8974538701x_{66} = 40752.8974538701
x67=38210.2363172014x_{67} = 38210.2363172014
x68=−41469.2292630202x_{68} = -41469.2292630202
x69=−32146.2318429411x_{69} = -32146.2318429411
x70=−23671.2234045161x_{70} = -23671.2234045161
x71=9399.14372710236x_{71} = 9399.14372710236
x72=−19434.0573719608x_{72} = -19434.0573719608
x73=24649.9050592789x_{73} = 24649.9050592789
x74=22954.9746878092x_{74} = 22954.9746878092
x75=−42316.7886675765x_{75} = -42316.7886675765
x76=−20281.4619905702x_{76} = -20281.4619905702
x77=−17739.3053561139x_{77} = -17739.3053561139
x78=−22823.7647944847x_{78} = -22823.7647944847
x79=29734.8864187221x_{79} = 29734.8864187221
x80=15328.5492029214x_{80} = 15328.5492029214
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 acot(x)/(1 + x^2).
acot⁡(0)02+1\frac{\operatorname{acot}{\left(0 \right)}}{0^{2} + 1}
Resultado:
f(0)=π2f{\left(0 \right)} = \frac{\pi}{2}
Punto:
(0, pi/2)
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
−2xacot⁡(x)(x2+1)2−1(x2+1)2=0- \frac{2 x \operatorname{acot}{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{1}{\left(x^{2} + 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=4162.02277941811x_{1} = 4162.02277941811
x2=−10015.0953632961x_{2} = -10015.0953632961
x3=−8270.69160497502x_{3} = -8270.69160497502
x4=7650.32380242809x_{4} = 7650.32380242809
x5=5470.0126744979x_{5} = 5470.0126744979
x6=8304.45720106198x_{6} = 8304.45720106198
x7=4815.98841973063x_{7} = 4815.98841973063
x8=−6744.40540599116x_{8} = -6744.40540599116
x9=1765.70397297034x_{9} = 1765.70397297034
x10=−5000.23190725497x_{10} = -5000.23190725497
x11=−6090.31427190719x_{11} = -6090.31427190719
x12=−7834.6016659414x_{12} = -7834.6016659414
x13=−8706.78660545704x_{13} = -8706.78660545704
x14=8740.55256141269x_{14} = 8740.55256141269
x15=−3038.56989473539x_{15} = -3038.56989473539
x16=1983.30578562056x_{16} = 1983.30578562056
x17=−10451.2045491196x_{17} = -10451.2045491196
x18=−6526.3723647221x_{18} = -6526.3723647221
x19=−3256.47459240601x_{19} = -3256.47459240601
x20=−7616.55886617619x_{20} = -7616.55886617619
x21=−2820.69079489299x_{21} = -2820.69079489299
x22=6996.20485915147x_{22} = 6996.20485915147
x23=4597.99212482363x_{23} = 4597.99212482363
x24=−3910.29945737163x_{24} = -3910.29945737163
x25=−1949.60219234621x_{25} = -1949.60219234621
x26=6124.07676110163x_{26} = 6124.07676110163
x27=8086.41134526104x_{27} = 8086.41134526104
x28=−8924.83577186377x_{28} = -8924.83577186377
x29=−6962.44077685779x_{29} = -6962.44077685779
x30=−4346.24722702185x_{30} = -4346.24722702185
x31=2636.57603133092x_{31} = 2636.57603133092
x32=3508.14854506539x_{32} = 3508.14854506539
x33=5906.05179014453x_{33} = 5906.05179014453
x34=−9142.8859430765x_{34} = -9142.8859430765
x35=−4782.23014302379x_{35} = -4782.23014302379
x36=−3692.34264418887x_{36} = -3692.34264418887
x37=9830.80846969027x_{37} = 9830.80846969027
x38=−8488.73852129304x_{38} = -8488.73852129304
x39=−5654.26888627018x_{39} = -5654.26888627018
x40=−7180.47826508317x_{40} = -7180.47826508317
x41=−9578.98902542867x_{41} = -9578.98902542867
x42=−10669.2601008739x_{42} = -10669.2601008739
x43=−4128.26825276073x_{43} = -4128.26825276073
x44=1548.23418175944x_{44} = 1548.23418175944
x45=−1732.01779022644x_{45} = -1732.01779022644
x46=9394.70345342889x_{46} = 9394.70345342889
x47=−2167.28435222334x_{47} = -2167.28435222334
x48=10484.9715229472x_{48} = 10484.9715229472
x49=1330.96129589603x_{49} = 1330.96129589603
x50=−10233.1496229718x_{50} = -10233.1496229718
x51=−5872.28981374269x_{51} = -5872.28981374269
x52=−5218.23939389624x_{52} = -5218.23939389624
x53=3944.05229914672x_{53} = 3944.05229914672
x54=−3474.4000691754x_{54} = -3474.4000691754
x55=9612.75555952255x_{55} = 9612.75555952255
x56=−10887.3162406434x_{56} = -10887.3162406434
x57=−9797.0418145963x_{57} = -9797.0418145963
x58=−2602.84372583742x_{58} = -2602.84372583742
x59=−4564.23492136622x_{59} = -4564.23492136622
x60=7214.24265801565x_{60} = 7214.24265801565
x61=5033.99112034038x_{61} = 5033.99112034038
x62=−7398.51768344902x_{62} = -7398.51768344902
x63=3726.09349528285x_{63} = 3726.09349528285
x64=10048.8621315856x_{64} = 10048.8621315856
x65=−8052.64595137848x_{65} = -8052.64595137848
x66=2418.76275647948x_{66} = 2418.76275647948
x67=4380.003192327x_{67} = 4380.003192327
x68=−1514.5733641732x_{68} = -1514.5733641732
x69=6560.13572998597x_{69} = 6560.13572998597
x70=7432.28236002073x_{70} = 7432.28236002073
x71=8958.60188833239x_{71} = 8958.60188833239
x72=9176.65220873118x_{72} = 9176.65220873118
x73=7868.36684053872x_{73} = 7868.36684053872
x74=−9360.93704887386x_{74} = -9360.93704887386
x75=−2385.03747217181x_{75} = -2385.03747217181
x76=6342.10484450584x_{76} = 6342.10484450584
x77=10703.0271681632x_{77} = 10703.0271681632
x78=2854.42856638056x_{78} = 2854.42856638056
x79=5688.03028959707x_{79} = 5688.03028959707
x80=8522.50430423249x_{80} = 8522.50430423249
x81=3072.31200448157x_{81} = 3072.31200448157
x82=6778.16914710792x_{82} = 6778.16914710792
x83=2201.00040928545x_{83} = 2201.00040928545
x84=3290.22020278613x_{84} = 3290.22020278613
x85=−1297.33949106901x_{85} = -1297.33949106901
x86=5251.99942880886x_{86} = 5251.99942880886
x87=10266.9164973095x_{87} = 10266.9164973095
x88=10921.0833958433x_{88} = 10921.0833958433
x89=−6308.34189463392x_{89} = -6308.34189463392
x90=−5436.25191436034x_{90} = -5436.25191436034
Signos de extremos en los puntos:
(4162.022779418108, 1.38703240658441e-11)

(-10015.095363296083, -9.95485015657526e-13)

(-8270.691604975023, -1.76756140325904e-12)

(7650.323802428094, 2.23336928202512e-12)

(5470.012674497899, 6.10991238728965e-12)

(8304.45720106198, 1.74608844362323e-12)

(4815.988419730635, 8.95248646577274e-12)

(-6744.4054059911605, -3.25963508581913e-12)

(1765.7039729703383, 1.81654487797078e-10)

(-5000.2319072549735, -7.9988865218599e-12)

(-6090.314271907193, -4.42670797977454e-12)

(-7834.601665941395, -2.07945357470403e-12)

(-8706.786605457044, -1.51504764592525e-12)

(8740.55256141269, 1.49755687781664e-12)

(-3038.5698947353944, -3.56444768741493e-11)

(1983.3057856205567, 1.28183113577173e-10)

(-10451.204549119557, -8.75993636708303e-13)

(-6526.3723647221, -3.59736445714227e-12)

(-3256.4745924060126, -2.89572198333679e-11)

(-7616.558866176195, -2.26320336988264e-12)

(-2820.6907948929857, -4.45587996919625e-11)

(6996.204859151467, 2.92019891221406e-12)

(4597.992124823627, 1.02871554682587e-11)

(-3910.2994573716264, -1.67251456734507e-11)

(-1949.602192346214, -1.34946564920017e-10)

(6124.076761101633, 4.35389656784e-12)

(8086.411345261038, 1.89117841531717e-12)

(-8924.835771863774, -1.40669289155061e-12)

(-6962.440776857787, -2.96288943856011e-12)

(-4346.247227021853, -1.21802636650844e-11)

(2636.5760313309215, 5.45605849402845e-11)

(3508.148545065389, 2.31614652202103e-11)

(5906.051790144531, 4.85409453929088e-12)

(-9142.885943076495, -1.30842877645413e-12)

(-4782.230143023788, -9.14341763477394e-12)

(-3692.3426441888723, -1.98652469892067e-11)

(9830.80846969027, 1.05252468233123e-12)

(-8488.738521293037, -1.63482218784145e-12)

(-5654.268886270178, -5.53185273230778e-12)

(-7180.478265083174, -2.70109508653999e-12)

(-9578.989025428667, -1.13773460226016e-12)

(-10669.260100873944, -8.23373882007275e-13)

(-4128.26825276073, -1.42133430015209e-11)

(1548.2341817594415, 2.6945720695752e-10)

(-1732.0177902264402, -1.92461010388805e-10)

(9394.703453428889, 1.20600960143814e-12)

(-2167.284352223341, -9.8231819992447e-11)

(10484.971522947208, 8.67557420775348e-13)

(1330.961295896035, 4.24134298249537e-10)

(-10233.149622971772, -9.33194180816615e-13)

(-5872.28981374269, -4.9383008000566e-12)

(-5218.239393896237, -7.03765513728496e-12)

(3944.0522991467187, 1.62994131160171e-11)

(-3474.4000691753954, -2.38429768455248e-11)

(9612.755559522555, 1.1257871756028e-12)

(-10887.31624064341, -7.74885405572818e-13)

(-9797.041814596298, -1.0634451844389e-12)

(-2602.843725837415, -5.67094757775458e-11)

(-4564.234921366225, -1.05171000101987e-11)

(7214.242658015648, 2.66334698352782e-12)

(5033.991120340382, 7.83903568976879e-12)

(-7398.51768344902, -2.46925442619765e-12)

(3726.0934952828516, 1.93303053415516e-11)

(10048.862131585647, 9.85483439891111e-13)

(-8052.645951378481, -1.91506789691464e-12)

(2418.762756479482, 7.06675577604318e-11)

(4380.003192326995, 1.19008145790019e-11)

(-1514.5733641732002, -2.87825191890463e-10)

(6560.135729985973, 3.54210565509135e-12)

(7432.282360020728, 2.4357537942883e-12)

(8958.601888332389, 1.39084675187226e-12)

(9176.652208731179, 1.29403843514829e-12)

(7868.366840538717, 2.05279788795437e-12)

(-9360.937048873859, -1.21910754572058e-12)

(-2385.0374721718117, -7.3707940713481e-11)

(6342.104844505838, 3.92012335952892e-12)

(10703.0271681632, 8.15605435601211e-13)

(2854.428566380563, 4.29974194780951e-11)

(5688.030289597067, 5.43393301973686e-12)

(8522.50430423249, 1.61546779391172e-12)

(3072.3120044815732, 3.44829160855005e-11)

(6778.169147107916, 3.21116617449959e-12)

(2201.0004092854515, 9.37863232915138e-11)

(3290.2202027861326, 2.80753418086247e-11)

(-1297.3394910690138, -4.57971804726728e-10)

(5251.99942880886, 6.90281077667208e-12)

(10266.91649730946, 9.24016879940413e-13)

(10921.083395843281, 7.6771994705811e-13)

(-6308.341894633916, -3.98340364926788e-12)

(-5436.251914360338, -6.22445396871597e-12)


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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico
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(3xx2+1+(4x2x2+1−1)acot⁡(x))(x2+1)2=0\frac{2 \left(\frac{3 x}{x^{2} + 1} + \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right) \operatorname{acot}{\left(x \right)}\right)}{\left(x^{2} + 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=2225.88482033618x_{1} = 2225.88482033618
x2=656.510598898059x_{2} = 656.510598898059
x3=−4244.41008366596x_{3} = -4244.41008366596
x4=2410.66431374473x_{4} = 2410.66431374473
x5=840.809186462159x_{5} = 840.809186462159
x6=−2581.14252154852x_{6} = -2581.14252154852
x7=2133.49864126527x_{7} = 2133.49864126527
x8=−1195.60096560461x_{8} = -1195.60096560461
x9=−1287.92239696698x_{9} = -1287.92239696698
x10=4628.36024254466x_{10} = 4628.36024254466
x11=1948.73511058234x_{11} = 1948.73511058234
x12=3519.4623685608x_{12} = 3519.4623685608
x13=−2211.57694900179x_{13} = -2211.57694900179
x14=−2303.96523664852x_{14} = -2303.96523664852
x15=1302.22125308858x_{15} = 1302.22125308858
x16=1856.3586380568x_{16} = 1856.3586380568
x17=−1842.05281317427x_{17} = -1842.05281317427
x18=−3597.55721889806x_{18} = -3597.55721889806
x19=2318.27347265262x_{19} = 2318.27347265262
x20=−2950.73395656248x_{20} = -2950.73395656248
x21=−4152.00124361308x_{21} = -4152.00124361308
x22=−4429.22880984772x_{22} = -4429.22880984772
x23=−642.252670828687x_{23} = -642.252670828687
x24=−550.255012112987x_{24} = -550.255012112987
x25=472.632985680155x_{25} = 472.632985680155
x26=2965.04386299706x_{26} = 2965.04386299706
x27=748.6267733249x_{27} = 748.6267733249
x28=1671.61920548606x_{28} = 1671.61920548606
x29=−2673.5383747917x_{29} = -2673.5383747917
x30=1579.25782731629x_{30} = 1579.25782731629
x31=−4336.8192798417x_{31} = -4336.8192798417
x32=2041.1152714347x_{32} = 2041.1152714347
x33=−3320.34261556465x_{33} = -3320.34261556465
x34=4166.31244361004x_{34} = 4166.31244361004
x35=564.493273032756x_{35} = 564.493273032756
x36=−1934.42866306114x_{36} = -1934.42866306114
x37=−2488.74825358209x_{37} = -2488.74825358209
x38=3796.68089593135x_{38} = 3796.68089593135
x39=4351.1305900676x_{39} = 4351.1305900676
x40=−458.427287084301x_{40} = -458.427287084301
x41=−1380.2566577561x_{41} = -1380.2566577561
x42=2872.6439568857x_{42} = 2872.6439568857
x43=−1564.95457442751x_{43} = -1564.95457442751
x44=3889.08811921794x_{44} = 3889.08811921794
x45=−4614.04879111071x_{45} = -4614.04879111071
x46=933.038147623406x_{46} = 933.038147623406
x47=3334.65307120044x_{47} = 3334.65307120044
x48=1025.30106538339x_{48} = 1025.30106538339
x49=−1103.29558948163x_{49} = -1103.29558948163
x50=−4521.63865321404x_{50} = -4521.63865321404
x51=4535.95006044959x_{51} = 4535.95006044959
x52=−3874.77711502024x_{52} = -3874.77711502024
x53=3242.24949549785x_{53} = 3242.24949549785
x54=−2119.19118318253x_{54} = -2119.19118318253
x55=−3043.13475177238x_{55} = -3043.13475177238
x56=3981.49580546587x_{56} = 3981.49580546587
x57=0.305668273602581x_{57} = 0.305668273602581
x58=−734.356067038591x_{58} = -734.356067038591
x59=−2396.35575432611x_{59} = -2396.35575432611
x60=3611.86798039772x_{60} = 3611.86798039772
x61=−918.752400461264x_{61} = -918.752400461264
x62=−1472.6013319921x_{62} = -1472.6013319921
x63=1209.89763934397x_{63} = 1209.89763934397
x64=3057.44481439486x_{64} = 3057.44481439486
x65=2687.84771184064x_{65} = 2687.84771184064
x66=1394.55727474212x_{66} = 1394.55727474212
x67=2595.45162696329x_{67} = 2595.45162696329
x68=−3412.74681562304x_{68} = -3412.74681562304
x69=2503.05710115595x_{69} = 2503.05710115595
x70=−2026.80828421394x_{70} = -2026.80828421394
x71=4073.9039231673x_{71} = 4073.9039231673
x72=−1657.31495142844x_{72} = -1657.31495142844
x73=−3135.53651383419x_{73} = -3135.53651383419
x74=3149.84671906841x_{74} = 3149.84671906841
x75=−3967.18473143275x_{75} = -3967.18473143275
x76=3427.05738151209x_{76} = 3427.05738151209
x77=−3689.96332176983x_{77} = -3689.96332176983
x78=−3227.93915969908x_{78} = -3227.93915969908
x79=1117.58951323931x_{79} = 1117.58951323931
x80=2780.24520030591x_{80} = 2780.24520030591
x81=−3782.36996674068x_{81} = -3782.36996674068
x82=−2765.93565439485x_{82} = -2765.93565439485
x83=1486.90339026949x_{83} = 1486.90339026949
x84=3704.27417025831x_{84} = 3704.27417025831
x85=1763.98643367305x_{85} = 1763.98643367305
x86=4258.72134057422x_{86} = 4258.72134057422
x87=−4059.59278400565x_{87} = -4059.59278400565
x88=−1011.01067450525x_{88} = -1011.01067450525
x89=−3505.15170100692x_{89} = -3505.15170100692
x90=−1749.68133227433x_{90} = -1749.68133227433
x91=−826.529713295631x_{91} = -826.529713295631
x92=4443.54017009392x_{92} = 4443.54017009392
x93=−2858.33422199463x_{93} = -2858.33422199463

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
[0.305668273602581,∞)\left[0.305668273602581, \infty\right)
Convexa en los intervalos
(−∞,0.305668273602581]\left(-\infty, 0.305668273602581\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(acot⁡(x)x2+1)=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 1}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(acot⁡(x)x2+1)=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 1}\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 acot(x)/(1 + x^2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(acot⁡(x)x(x2+1))=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x \left(x^{2} + 1\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(acot⁡(x)x(x2+1))=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x \left(x^{2} + 1\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:
acot⁡(x)x2+1=−acot⁡(x)x2+1\frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 1} = - \frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 1}
- No
acot⁡(x)x2+1=acot⁡(x)x2+1\frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 1} = \frac{\operatorname{acot}{\left(x \right)}}{x^{2} + 1}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = acot(x)/(1+x^2)
    © Sr Examen – Калькулятор