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((x-1)*arctan(1/x))/(x^2-3x+2)

  • ¿Cómo usar?

  • Gráfico de la función y =:
  • 4*x-x^2 4*x-x^2
  • 2*x^3-15*x^2+36*x-32 2*x^3-15*x^2+36*x-32
  • (x+4)/e^(x+4) (x+4)/e^(x+4)
  • x^3+4*x x^3+4*x

  • Expresiones idénticas

  • ((x- uno)*arctan(uno /x))/(x^ dos -3x+ dos)
  • ((x menos 1) multiplicar por arc tangente de (1 dividir por x)) dividir por (x al cuadrado menos 3x más 2)
  • ((x menos uno) multiplicar por arc tangente de (uno dividir por x)) dividir por (x en el grado dos menos 3x más dos)
  • ((x-1)*arctan(1/x))/(x2-3x+2)
  • x-1*arctan1/x/x2-3x+2
  • ((x-1)*arctan(1/x))/(x²-3x+2)
  • ((x-1)*arctan(1/x))/(x en el grado 2-3x+2)
  • ((x-1)arctan(1/x))/(x^2-3x+2)
  • ((x-1)arctan(1/x))/(x2-3x+2)
  • x-1arctan1/x/x2-3x+2
  • x-1arctan1/x/x^2-3x+2
  • ((x-1)*arctan(1 dividir por x)) dividir por (x^2-3x+2)

  • Expresiones semejantes

  • ((x+1)*arctan(1/x))/(x^2-3x+2)
  • ((x-1)*arctan(1/x))/(x^2+3x+2)
  • ((x-1)*arctan(1/x))/(x^2-3x-2)

  • Expresiones con funciones

  • arctan
  • arctan(e^(1/x))
Trazado el gráfico de la función/ ((x-1)*arctan(1/x))/(x^2-3x+2)

Gráfico de la función y = ((x-1)*arctan(1/x))/(x^2-3x+2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                   /1\
       (x - 1)*atan|-|
                   \x/
f(x) = ---------------
          2           
         x  - 3*x + 2
f(x)=(x−1)atan⁡(1x)(x2−3x)+2f{\left(x \right)} = \frac{\left(x - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x^{2} - 3 x\right) + 2} 
f = ((x - 1)*atan(1/x))/(x^2 - 3*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=1x_{2} = 1
x3=2x_{3} = 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:
(x−1)atan⁡(1x)(x2−3x)+2=0\frac{\left(x - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x^{2} - 3 x\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 ((x - 1)*atan(1/x))/(x^2 - 3*x + 2).
(−1)atan⁡(10)(02−0)+2\frac{\left(-1\right) \operatorname{atan}{\left(\frac{1}{0} \right)}}{\left(0^{2} - 0\right) + 2}
Resultado:
f(0)=⟨−π4,π4⟩f{\left(0 \right)} = \left\langle - \frac{\pi}{4}, \frac{\pi}{4}\right\rangle
Punto:
(0, AccumBounds(-pi/4, pi/4))
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
(3−2x)(x−1)atan⁡(1x)((x2−3x)+2)2+atan⁡(1x)−x−1x2(1+1x2)(x2−3x)+2=0\frac{\left(3 - 2 x\right) \left(x - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(\left(x^{2} - 3 x\right) + 2\right)^{2}} + \frac{\operatorname{atan}{\left(\frac{1}{x} \right)} - \frac{x - 1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{\left(x^{2} - 3 x\right) + 2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=19132.901290678x_{1} = 19132.901290678
x2=−31715.782844909x_{2} = -31715.782844909
x3=22524.0604197956x_{3} = 22524.0604197956
x4=−20695.5049116438x_{4} = -20695.5049116438
x5=26762.7395638698x_{5} = 26762.7395638698
x6=37782.6044575101x_{6} = 37782.6044575101
x7=41173.2218900857x_{7} = 41173.2218900857
x8=−25781.934144489x_{8} = -25781.934144489
x9=39477.9175405522x_{9} = 39477.9175405522
x10=−34258.7965114625x_{10} = -34258.7965114625
x11=21676.2929417899x_{11} = 21676.2929417899
x12=27610.4506732461x_{12} = 27610.4506732461
x13=42020.871124463x_{13} = 42020.871124463
x14=−27477.346095126x_{14} = -27477.346095126
x15=−36801.7829103825x_{15} = -36801.7829103825
x16=32696.5989933765x_{16} = 32696.5989933765
x17=36934.9442594904x_{17} = 36934.9442594904
x18=40325.5707372207x_{18} = 40325.5707372207
x19=−29172.7349491019x_{19} = -29172.7349491019
x20=−13064.7511312679x_{20} = -13064.7511312679
x21=34391.9468419258x_{21} = 34391.9468419258
x22=−39344.7472362699x_{22} = -39344.7472362699
x23=−32563.4574828023x_{23} = -32563.4574828023
x24=16589.3264918846x_{24} = 16589.3264918846
x25=13197.4187357505x_{25} = 13197.4187357505
x26=15741.4117020698x_{26} = 15741.4117020698
x27=−38497.0946483211x_{27} = -38497.0946483211
x28=31848.9194068878x_{28} = 31848.9194068878
x29=25915.0213145912x_{29} = 25915.0213145912
x30=28458.1552925966x_{30} = 28458.1552925966
x31=29305.8539952984x_{31} = 29305.8539952984
x32=23371.8157836424x_{32} = 23371.8157836424
x33=35239.6156571822x_{33} = 35239.6156571822
x34=−30020.4219087624x_{34} = -30020.4219087624
x35=33544.2747081237x_{35} = 33544.2747081237
x36=−30868.1044245516x_{36} = -30868.1044245516
x37=−33411.1286201263x_{37} = -33411.1286201263
x38=−14760.6707477129x_{38} = -14760.6707477129
x39=20828.5118341179x_{39} = 20828.5118341179
x40=−41887.6934436303x_{40} = -41887.6934436303
x41=14045.4655779585x_{41} = 14045.4655779585
x42=−37649.4399213803x_{42} = -37649.4399213803
x43=−35106.4613873068x_{43} = -35106.4613873068
x44=25067.2951858459x_{44} = 25067.2951858459
x45=14893.4603458192x_{45} = 14893.4603458192
x46=−18999.9363839473x_{46} = -18999.9363839473
x47=30153.5472891576x_{47} = 30153.5472891576
x48=−24934.2180776959x_{48} = -24934.2180776959
x49=18285.0672504789x_{49} = 18285.0672504789
x50=−18152.1278863196x_{50} = -18152.1278863196
x51=−17304.3002060059x_{51} = -17304.3002060059
x52=36087.2813912772x_{52} = 36087.2813912772
x53=−21543.2686222863x_{53} = -21543.2686222863
x54=−23238.7621938086x_{54} = -23238.7621938086
x55=31001.235625517x_{55} = 31001.235625517
x56=−15608.5754127483x_{56} = -15608.5754127483
x57=−41040.046516552x_{57} = -41040.046516552
x58=−35954.12345685x_{58} = -35954.12345685
x59=−40192.3978182678x_{59} = -40192.3978182678
x60=−24086.4943643894x_{60} = -24086.4943643894
x61=−19847.7280620064x_{61} = -19847.7280620064
x62=24219.5603323716x_{62} = 24219.5603323716
x63=17437.2101951941x_{63} = 17437.2101951941
x64=−42735.3387032293x_{64} = -42735.3387032293
x65=38630.2621636119x_{65} = 38630.2621636119
x66=−16456.4505171738x_{66} = -16456.4505171738
x67=−12216.721901123x_{67} = -12216.721901123
x68=−28325.043156493x_{68} = -28325.043156493
x69=−13912.7314292036x_{69} = -13912.7314292036
x70=−22391.0206365865x_{70} = -22391.0206365865
x71=−26629.6432752343x_{71} = -26629.6432752343
x72=19980.7153178136x_{72} = 19980.7153178136
x73=12349.3086944618x_{73} = 12349.3086944618
Signos de extremos en los puntos:
(19132.90129067801, 2.73201914114813e-9)

(-31715.782844908965, 9.94080930629393e-10)

(22524.060419795554, 1.97126584340574e-9)

(-20695.504911643824, 2.33456504655437e-9)

(26762.739563869836, 1.39627615196543e-9)

(37782.604457510126, 7.00550110966409e-10)

(41173.22189008572, 5.89917650874476e-10)

(-25781.93414448899, 1.50430300059396e-9)

(39477.9175405522, 6.41672653614552e-10)

(-34258.79651146253, 8.51982044306463e-10)

(21676.292941789852, 2.12848233050144e-9)

(27610.450673246065, 1.31185087306286e-9)

(42020.87112446296, 5.66357385184307e-10)

(-27477.346095126024, 1.32439893203344e-9)

(-36801.78291038248, 7.38309880462269e-10)

(32696.59899337655, 9.35451776124229e-10)

(36934.944259490374, 7.33075358403515e-10)

(40325.57073722067, 6.14979295029748e-10)

(-29172.734949101923, 1.17494066314519e-9)

(-13064.751131267925, 5.85775546935869e-9)

(34391.94684192579, 8.45496344998119e-10)

(-39344.74723626989, 6.45958183965234e-10)

(-32563.457482802325, 9.43001330834122e-10)

(16589.326491884636, 3.63408308226448e-9)

(13197.418735750478, 5.74232576061641e-9)

(15741.411702069821, 4.0361549088701e-9)

(-38497.09464832114, 6.74716806952943e-10)

(31848.919406887762, 9.85911356411611e-10)

(25915.02131459115, 1.48912233357487e-9)

(28458.15529259655, 1.2348580369536e-9)

(29305.853995298414, 1.16445008370026e-9)

(23371.815783642378, 1.83084785758887e-9)

(35239.61565718223, 8.05308573002602e-10)

(-30020.421908762426, 1.10952600188143e-9)

(33544.27470812366, 8.88769377075685e-10)

(-30868.10442455163, 1.0494262915831e-9)

(-33411.12862012634, 8.95760094770639e-10)

(-14760.670747712868, 4.58911533456979e-9)

(20828.511834117886, 2.30528816593101e-9)

(-41887.69344363028, 5.69910123169637e-10)

(14045.465577958506, 5.06978527224611e-9)

(-37649.43992138033, 7.05439689234011e-10)

(-35106.461387306765, 8.1133674709857e-10)

(25067.295185845906, 1.59154785294615e-9)

(14893.460345819176, 4.50886359085526e-9)

(-18999.936383947294, 2.76981008975357e-9)

(30153.547289157617, 1.09989691975706e-9)

(-24934.218077695885, 1.60832443013563e-9)

(18285.067250478853, 2.99126149718981e-9)

(-18152.12788631959, 3.03456934614772e-9)

(-17304.300206005853, 3.33919457088177e-9)

(36087.28139127721, 7.67919573740742e-10)

(-21543.26862228632, 2.15445034881801e-9)

(-23238.762193808594, 1.85155510188769e-9)

(31001.235625516987, 1.04056690865846e-9)

(-15608.575412748307, 4.10409892781005e-9)

(-41040.04651655202, 5.93694666328046e-10)

(-35954.12345684996, 7.7353226186624e-10)

(-40192.397818267775, 6.1899986887773e-10)

(-24086.494364389437, 1.7235216707421e-9)

(-19847.728062006405, 2.5382514170452e-9)

(24219.56033237159, 1.70491744455607e-9)

(17437.21019519413, 3.28924186510581e-9)

(-42735.33870322935, 5.4752679147855e-10)

(38630.26216361191, 6.70142519606563e-10)

(-16456.45051717375, 3.69211215543147e-9)

(-12216.721901122954, 6.69914735818198e-9)

(-28325.043156493033, 1.24631601412484e-9)

(-13912.731429203583, 5.16550479816188e-9)

(-22391.020636586494, 1.99440534389897e-9)

(-26629.643275234255, 1.41005705594739e-9)

(19980.715317813643, 2.505078900208e-9)

(12349.308694461808, 6.55820587966886e-9)


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((x−1)((2x−3)2x2−3x+2−1)atan⁡(1x)x2−3x+2−(2x−3)(atan⁡(1x)−x−1x2(1+1x2))x2−3x+2−1−(1−1x2(1+1x2))(x−1)xx2(1+1x2))x2−3x+2=0\frac{2 \left(\frac{\left(x - 1\right) \left(\frac{\left(2 x - 3\right)^{2}}{x^{2} - 3 x + 2} - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{x^{2} - 3 x + 2} - \frac{\left(2 x - 3\right) \left(\operatorname{atan}{\left(\frac{1}{x} \right)} - \frac{x - 1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2} - \frac{1 - \frac{\left(1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right) \left(x - 1\right)}{x}}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=10157.4333881273x_{1} = 10157.4333881273
x2=−10775.9395495359x_{2} = -10775.9395495359
x3=1867.3087499301x_{3} = 1867.3087499301
x4=−2705.54320615931x_{4} = -2705.54320615931
x5=3177.5277208136x_{5} = 3177.5277208136
x6=−3578.38429227779x_{6} = -3578.38429227779
x7=−8376.98463267766x_{7} = -8376.98463267766
x8=8630.81238261251x_{8} = 8630.81238261251
x9=4050.34034635558x_{9} = 4050.34034635558
x10=−3141.99957042143x_{10} = -3141.99957042143
x11=−3796.55694031468x_{11} = -3796.55694031468
x12=2304.28790241106x_{12} = 2304.28790241106
x13=−2923.78215547567x_{13} = -2923.78215547567
x14=−5541.66011870512x_{14} = -5541.66011870512
x15=−8813.16546743902x_{15} = -8813.16546743902
x16=5795.47301832209x_{16} = 5795.47301832209
x17=−3360.19928749126x_{17} = -3360.19928749126
x18=3613.9674167136x_{18} = 3613.9674167136
x19=6667.92469654352x_{19} = 6667.92469654352
x20=−7504.60927199796x_{20} = -7504.60927199796
x21=4486.66656950272x_{21} = 4486.66656950272
x22=7322.23616233156x_{22} = 7322.23616233156
x23=9066.99434174586x_{23} = 9066.99434174586
x24=−4014.71911340662x_{24} = -4014.71911340662
x25=10375.5189441485x_{25} = 10375.5189441485
x26=3395.75758558922x_{26} = 3395.75758558922
x27=2522.66333071821x_{27} = 2522.66333071821
x28=−9903.60231911807x_{28} = -9903.60231911807
x29=−6195.99805407436x_{29} = -6195.99805407436
x30=5359.22570745292x_{30} = 5359.22570745292
x31=10811.688172078x_{31} = 10811.688172078
x32=7104.13445049151x_{32} = 7104.13445049151
x33=8194.62591523746x_{33} = 8194.62591523746
x34=7976.53075517285x_{34} = 7976.53075517285
x35=−9467.42989616817x_{35} = -9467.42989616817
x36=−6850.31244807713x_{36} = -6850.31244807713
x37=1648.64082061498x_{37} = 1648.64082061498
x38=6231.70498831242x_{38} = 6231.70498831242
x39=−4232.87233209159x_{39} = -4232.87233209159
x40=−10557.8560975106x_{40} = -10557.8560975106
x41=−10121.6875184064x_{41} = -10121.6875184064
x42=−8158.89263749768x_{42} = -8158.89263749768
x43=−4887.28963035618x_{43} = -4887.28963035618
x44=2740.98786243025x_{44} = 2740.98786243025
x45=7758.43416347178x_{45} = 7758.43416347178
x46=−7068.41310827937x_{46} = -7068.41310827937
x47=−9685.5164599013x_{47} = -9685.5164599013
x48=−1832.24305595666x_{48} = -1832.24305595666
x49=−7940.79947803707x_{49} = -7940.79947803707
x50=5141.0950432625x_{50} = 5141.0950432625
x51=−5977.88839183217x_{51} = -5977.88839183217
x52=−9031.2544557902x_{52} = -9031.2544557902
x53=−7286.51199970116x_{53} = -7286.51199970116
x54=5577.35151153877x_{54} = 5577.35151153877
x55=−6414.10512463753x_{55} = -6414.10512463753
x56=−8595.07554997102x_{56} = -8595.07554997102
x57=4922.95885359251x_{57} = 4922.95885359251
x58=6013.59070704205x_{58} = 6013.59070704205
x59=−4451.01783778936x_{59} = -4451.01783778936
x60=−4669.15665366985x_{60} = -4669.15665366985
x61=−2487.2777178632x_{61} = -2487.2777178632
x62=−5323.54080632076x_{62} = -5323.54080632076
x63=4704.81634605761x_{63} = 4704.81634605761
x64=−10994.0224866691x_{64} = -10994.0224866691
x65=−7722.70505842844x_{65} = -7722.70505842844
x66=8412.71975712246x_{66} = 8412.71975712246
x67=10593.603859001x_{67} = 10593.603859001
x68=9721.2601770401x_{68} = 9721.2601770401
x69=9939.34714803107x_{69} = 9939.34714803107
x70=2085.84474314979x_{70} = 2085.84474314979
x71=3832.16075254444x_{71} = 3832.16075254444
x72=−10339.772098672x_{72} = -10339.772098672
x73=−1613.77630220934x_{73} = -1613.77630220934
x74=9285.0838327236x_{74} = 9285.0838327236
x75=4268.50837223294x_{75} = 4268.50837223294
x76=2959.27319884515x_{76} = 2959.27319884515
x77=−5105.41748066561x_{77} = -5105.41748066561
x78=−2268.97905475936x_{78} = -2268.97905475936
x79=−9249.34257922523x_{79} = -9249.34257922523
x80=6886.03069691095x_{80} = 6886.03069691095
x81=7540.33601328405x_{81} = 7540.33601328405
x82=−5759.77585515002x_{82} = -5759.77585515002
x83=−2050.63826429131x_{83} = -2050.63826429131
x84=8848.90388328254x_{84} = 8848.90388328254
x85=9503.17242396468x_{85} = 9503.17242396468
x86=6449.81621596843x_{86} = 6449.81621596843
x87=−6632.20985044559x_{87} = -6632.20985044559
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=1x_{2} = 1
x3=2x_{3} = 2

lim⁡x→0−(2((x−1)((2x−3)2x2−3x+2−1)atan⁡(1x)x2−3x+2−(2x−3)(atan⁡(1x)−x−1x2(1+1x2))x2−3x+2−1−(1−1x2(1+1x2))(x−1)xx2(1+1x2))x2−3x+2)=π8+12\lim_{x \to 0^-}\left(\frac{2 \left(\frac{\left(x - 1\right) \left(\frac{\left(2 x - 3\right)^{2}}{x^{2} - 3 x + 2} - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{x^{2} - 3 x + 2} - \frac{\left(2 x - 3\right) \left(\operatorname{atan}{\left(\frac{1}{x} \right)} - \frac{x - 1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2} - \frac{1 - \frac{\left(1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right) \left(x - 1\right)}{x}}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2}\right) = \frac{\pi}{8} + \frac{1}{2}
lim⁡x→0+(2((x−1)((2x−3)2x2−3x+2−1)atan⁡(1x)x2−3x+2−(2x−3)(atan⁡(1x)−x−1x2(1+1x2))x2−3x+2−1−(1−1x2(1+1x2))(x−1)xx2(1+1x2))x2−3x+2)=12−π8\lim_{x \to 0^+}\left(\frac{2 \left(\frac{\left(x - 1\right) \left(\frac{\left(2 x - 3\right)^{2}}{x^{2} - 3 x + 2} - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{x^{2} - 3 x + 2} - \frac{\left(2 x - 3\right) \left(\operatorname{atan}{\left(\frac{1}{x} \right)} - \frac{x - 1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2} - \frac{1 - \frac{\left(1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right) \left(x - 1\right)}{x}}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2}\right) = \frac{1}{2} - \frac{\pi}{8}
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión
lim⁡x→1−(2((x−1)((2x−3)2x2−3x+2−1)atan⁡(1x)x2−3x+2−(2x−3)(atan⁡(1x)−x−1x2(1+1x2))x2−3x+2−1−(1−1x2(1+1x2))(x−1)xx2(1+1x2))x2−3x+2)=−1.0707963267949\lim_{x \to 1^-}\left(\frac{2 \left(\frac{\left(x - 1\right) \left(\frac{\left(2 x - 3\right)^{2}}{x^{2} - 3 x + 2} - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{x^{2} - 3 x + 2} - \frac{\left(2 x - 3\right) \left(\operatorname{atan}{\left(\frac{1}{x} \right)} - \frac{x - 1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2} - \frac{1 - \frac{\left(1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right) \left(x - 1\right)}{x}}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2}\right) = -1.0707963267949
lim⁡x→1+(2((x−1)((2x−3)2x2−3x+2−1)atan⁡(1x)x2−3x+2−(2x−3)(atan⁡(1x)−x−1x2(1+1x2))x2−3x+2−1−(1−1x2(1+1x2))(x−1)xx2(1+1x2))x2−3x+2)=−1.0707963267949\lim_{x \to 1^+}\left(\frac{2 \left(\frac{\left(x - 1\right) \left(\frac{\left(2 x - 3\right)^{2}}{x^{2} - 3 x + 2} - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{x^{2} - 3 x + 2} - \frac{\left(2 x - 3\right) \left(\operatorname{atan}{\left(\frac{1}{x} \right)} - \frac{x - 1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2} - \frac{1 - \frac{\left(1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right) \left(x - 1\right)}{x}}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2}\right) = -1.0707963267949
- los límites son iguales, es decir omitimos el punto correspondiente
lim⁡x→2−(2((x−1)((2x−3)2x2−3x+2−1)atan⁡(1x)x2−3x+2−(2x−3)(atan⁡(1x)−x−1x2(1+1x2))x2−3x+2−1−(1−1x2(1+1x2))(x−1)xx2(1+1x2))x2−3x+2)=−∞\lim_{x \to 2^-}\left(\frac{2 \left(\frac{\left(x - 1\right) \left(\frac{\left(2 x - 3\right)^{2}}{x^{2} - 3 x + 2} - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{x^{2} - 3 x + 2} - \frac{\left(2 x - 3\right) \left(\operatorname{atan}{\left(\frac{1}{x} \right)} - \frac{x - 1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2} - \frac{1 - \frac{\left(1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right) \left(x - 1\right)}{x}}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2}\right) = -\infty
lim⁡x→2+(2((x−1)((2x−3)2x2−3x+2−1)atan⁡(1x)x2−3x+2−(2x−3)(atan⁡(1x)−x−1x2(1+1x2))x2−3x+2−1−(1−1x2(1+1x2))(x−1)xx2(1+1x2))x2−3x+2)=∞\lim_{x \to 2^+}\left(\frac{2 \left(\frac{\left(x - 1\right) \left(\frac{\left(2 x - 3\right)^{2}}{x^{2} - 3 x + 2} - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{x^{2} - 3 x + 2} - \frac{\left(2 x - 3\right) \left(\operatorname{atan}{\left(\frac{1}{x} \right)} - \frac{x - 1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2} - \frac{1 - \frac{\left(1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right) \left(x - 1\right)}{x}}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{2} - 3 x + 2}\right) = \infty
- los límites no son iguales, signo
x3=2x_{3} = 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:
No tiene corvaduras en todo el eje numérico
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=1x_{2} = 1
x3=2x_{3} = 2
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((x−1)atan⁡(1x)(x2−3x)+2)=0\lim_{x \to -\infty}\left(\frac{\left(x - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x^{2} - 3 x\right) + 2}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((x−1)atan⁡(1x)(x2−3x)+2)=0\lim_{x \to \infty}\left(\frac{\left(x - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x^{2} - 3 x\right) + 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 ((x - 1)*atan(1/x))/(x^2 - 3*x + 2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((x−1)atan⁡(1x)x((x2−3x)+2))=0\lim_{x \to -\infty}\left(\frac{\left(x - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{x \left(\left(x^{2} - 3 x\right) + 2\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((x−1)atan⁡(1x)x((x2−3x)+2))=0\lim_{x \to \infty}\left(\frac{\left(x - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{x \left(\left(x^{2} - 3 x\right) + 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:
(x−1)atan⁡(1x)(x2−3x)+2=−(−x−1)atan⁡(1x)x2+3x+2\frac{\left(x - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x^{2} - 3 x\right) + 2} = - \frac{\left(- x - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{x^{2} + 3 x + 2}
- No
(x−1)atan⁡(1x)(x2−3x)+2=(−x−1)atan⁡(1x)x2+3x+2\frac{\left(x - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x^{2} - 3 x\right) + 2} = \frac{\left(- x - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{x^{2} + 3 x + 2}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = ((x-1)*arctan(1/x))/(x^2-3x+2)
    © Sr Examen – Калькулятор