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(1+cos(x))/x
Trazado el gráfico de la función/ (1+cos(x))/x

Gráfico de la función y = (1+cos(x))/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       1 + cos(x)
f(x) = ----------
           x
f(x)=cos⁡(x)+1xf{\left(x \right)} = \frac{\cos{\left(x \right)} + 1}{x} 
f = (cos(x) + 1)/x
Gráfico de la función
02468-8-6-4-2-1010-100100
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
cos⁡(x)+1x=0\frac{\cos{\left(x \right)} + 1}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=πx_{1} = \pi
Solución numérica
x1=−15.7079633360896x_{1} = -15.7079633360896
x2=−91.1061864740646x_{2} = -91.1061864740646
x3=−40.8407048905025x_{3} = -40.8407048905025
x4=−97.3893724524824x_{4} = -97.3893724524824
x5=9.42477824522119x_{5} = 9.42477824522119
x6=−28.2743343549045x_{6} = -28.2743343549045
x7=15.7079634470871x_{7} = 15.7079634470871
x8=−28.2743338085311x_{8} = -28.2743338085311
x9=3.14159295485917x_{9} = 3.14159295485917
x10=−97.3893717372023x_{10} = -97.3893717372023
x11=72.2566310277169x_{11} = 72.2566310277169
x12=−59.6902604577438x_{12} = -59.6902604577438
x13=958.185757046208x_{13} = 958.185757046208
x14=40.8407049627545x_{14} = 40.8407049627545
x15=−34.5575203552595x_{15} = -34.5575203552595
x16=−97.3893714809928x_{16} = -97.3893714809928
x17=−34.5575196459487x_{17} = -34.5575196459487
x18=53.4070754202298x_{18} = 53.4070754202298
x19=78.5398168423732x_{19} = 78.5398168423732
x20=−91.1061872629307x_{20} = -91.1061872629307
x21=116.238911986049x_{21} = 116.238911986049
x22=47.123889402027x_{22} = 47.123889402027
x23=−21.9911480930588x_{23} = -21.9911480930588
x24=9.42477740376207x_{24} = 9.42477740376207
x25=78.5398164979331x_{25} = 78.5398164979331
x26=−53.4070743239768x_{26} = -53.4070743239768
x27=84.8230021249244x_{27} = 84.8230021249244
x28=−21.9911489783129x_{28} = -21.9911489783129
x29=−65.9734461708474x_{29} = -65.9734461708474
x30=15.7079622060617x_{30} = 15.7079622060617
x31=59.6902602175539x_{31} = 59.6902602175539
x32=−53.4070745765549x_{32} = -53.4070745765549
x33=65.9734459897915x_{33} = 65.9734459897915
x34=72.256630656682x_{34} = 72.256630656682
x35=−21.991148586429x_{35} = -21.991148586429
x36=−28.2743337044571x_{36} = -28.2743337044571
x37=−47.1238893125088x_{37} = -47.1238893125088
x38=40.8407042012327x_{38} = 40.8407042012327
x39=−3.14159289587022x_{39} = -3.14159289587022
x40=65.973445752956x_{40} = 65.973445752956
x41=−84.8230020513102x_{41} = -84.8230020513102
x42=−65.973445764876x_{42} = -65.973445764876
x43=−47.123890103677x_{43} = -47.123890103677
x44=21.9911480146302x_{44} = 21.9911480146302
x45=−59.6902598948281x_{45} = -59.6902598948281
x46=72.2566314937548x_{46} = 72.2566314937548
x47=−47.1238897845322x_{47} = -47.1238897845322
x48=21.9911504378094x_{48} = 21.9911504378094
x49=15.7079626859982x_{49} = 15.7079626859982
x50=−72.2566310742307x_{50} = -72.2566310742307
x51=21.9911472569555x_{51} = 21.9911472569555
x52=97.3893717891282x_{52} = 97.3893717891282
x53=−59.6902606318082x_{53} = -59.6902606318082
x54=21.9911501136688x_{54} = 21.9911501136688
x55=−78.5398160446852x_{55} = -78.5398160446852
x56=91.1061865624362x_{56} = 91.1061865624362
x57=−197.920329199208x_{57} = -197.920329199208
x58=97.3893725791873x_{58} = 97.3893725791873
x59=−72.2566315269131x_{59} = -72.2566315269131
x60=−97.3893744406124x_{60} = -97.3893744406124
x61=21.9911485851958x_{61} = 21.9911485851958
x62=47.1238902068628x_{62} = 47.1238902068628
x63=53.4070746289822x_{63} = 53.4070746289822
x64=−91.1061868565768x_{64} = -91.1061868565768
x65=−84.8230012463222x_{65} = -84.8230012463222
x66=−15.7079626645131x_{66} = -15.7079626645131
x67=28.2743343156322x_{67} = 28.2743343156322
x68=−65.9734453187365x_{68} = -65.9734453187365
x69=91.1061873667666x_{69} = 91.1061873667666
x70=−40.8407040846935x_{70} = -40.8407040846935
x71=−15.7079632964851x_{71} = -15.7079632964851
x72=34.5575190200753x_{72} = 34.5575190200753
x73=179.070787703886x_{73} = 179.070787703886
x74=−72.2566308649321x_{74} = -72.2566308649321
x75=−59.6902609730457x_{75} = -59.6902609730457
x76=−3.14159179672595x_{76} = -3.14159179672595
x77=−9.42477731294084x_{77} = -9.42477731294084
x78=−78.5398169122209x_{78} = -78.5398169122209
x79=91.1061863403424x_{79} = 91.1061863403424
x80=−78.5398168102871x_{80} = -78.5398168102871
x81=28.274333454086x_{81} = 28.274333454086
x82=34.5575192607047x_{82} = 34.5575192607047
x83=78.5398161797687x_{83} = 78.5398161797687
x84=59.6902598603332x_{84} = 59.6902598603332
x85=−34.5575188836103x_{85} = -34.5575188836103
x86=65.9734452145909x_{86} = 65.9734452145909
x87=−9.42477812949905x_{87} = -9.42477812949905
x88=84.8230013613497x_{88} = 84.8230013613497
x89=34.5575196754943x_{89} = 34.5575196754943
x90=21.9911487653177x_{90} = 21.9911487653177
x91=28.2743338651569x_{91} = 28.2743338651569
x92=3.14159206272542x_{92} = 3.14159206272542
x93=59.6902606090451x_{93} = 59.6902606090451
x94=−53.4070752935774x_{94} = -53.4070752935774
x95=59.6902598928167x_{95} = 59.6902598928167
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 (1 + cos(x))/x.
1+cos⁡(0)0\frac{1 + \cos{\left(0 \right)}}{0}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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
−sin⁡(x)x−cos⁡(x)+1x2=0- \frac{\sin{\left(x \right)}}{x} - \frac{\cos{\left(x \right)} + 1}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−50.2256674407532x_{1} = -50.2256674407532
x2=−87.941852980689x_{2} = -87.941852980689
x3=25.0529526753384x_{3} = 25.0529526753384
x4=84.8230016469244x_{4} = 84.8230016469244
x5=47.1238898038469x_{5} = 47.1238898038469
x6=100.511067265113x_{6} = 100.511067265113
x7=−53.4070751110265x_{7} = -53.4070751110265
x8=−37.6459978360151x_{8} = -37.6459978360151
x9=91.106186954104x_{9} = 91.106186954104
x10=−84.8230016469244x_{10} = -84.8230016469244
x11=5.95017264337656x_{11} = 5.95017264337656
x12=−3.14159265358979x_{12} = -3.14159265358979
x13=−69.086091013299x_{13} = -69.086091013299
x14=87.941852980689x_{14} = 87.941852980689
x15=−279.601746169492x_{15} = -279.601746169492
x16=−40.8407044966673x_{16} = -40.8407044966673
x17=62.8000086337252x_{17} = 62.8000086337252
x18=−62.8000086337252x_{18} = -62.8000086337252
x19=−100.511067265113x_{19} = -100.511067265113
x20=−12.4054996335861x_{20} = -12.4054996335861
x21=78.5398163397448x_{21} = 78.5398163397448
x22=477.51789502706x_{22} = 477.51789502706
x23=−18.7429502117119x_{23} = -18.7429502117119
x24=−9.42477796076938x_{24} = -9.42477796076938
x25=94.2265549654551x_{25} = 94.2265549654551
x26=72.2566310325652x_{26} = 72.2566310325652
x27=56.5132815466599x_{27} = 56.5132815466599
x28=9.42477796076938x_{28} = 9.42477796076938
x29=43.9367850637406x_{29} = 43.9367850637406
x30=40.8407044966673x_{30} = 40.8407044966673
x31=59.6902604182061x_{31} = 59.6902604182061
x32=−25.0529526753384x_{32} = -25.0529526753384
x33=−91.106186954104x_{33} = -91.106186954104
x34=97.3893722612836x_{34} = 97.3893722612836
x35=34.5575191894877x_{35} = 34.5575191894877
x36=75.3716900810604x_{36} = 75.3716900810604
x37=−78.5398163397448x_{37} = -78.5398163397448
x38=37.6459978360151x_{38} = 37.6459978360151
x39=31.3521566903887x_{39} = 31.3521566903887
x40=779.112411068009x_{40} = 779.112411068009
x41=81.6569174978428x_{41} = 81.6569174978428
x42=−65.9734457253857x_{42} = -65.9734457253857
x43=−43.9367850637406x_{43} = -43.9367850637406
x44=12.4054996335861x_{44} = 12.4054996335861
x45=3.14159265358979x_{45} = 3.14159265358979
x46=15.707963267949x_{46} = 15.707963267949
x47=−31.3521566903887x_{47} = -31.3521566903887
x48=−21.9911485751286x_{48} = -21.9911485751286
x49=18.7429502117119x_{49} = 18.7429502117119
x50=−15.707963267949x_{50} = -15.707963267949
x51=69.086091013299x_{51} = 69.086091013299
x52=128.805298797182x_{52} = 128.805298797182
x53=−94.2265549654551x_{53} = -94.2265549654551
x54=28.2743338823081x_{54} = 28.2743338823081
x55=−59.6902604182061x_{55} = -59.6902604182061
x56=−408.402147842567x_{56} = -408.402147842567
x57=−81.6569174978428x_{57} = -81.6569174978428
x58=−97.3893722612836x_{58} = -97.3893722612836
x59=21.9911485751286x_{59} = 21.9911485751286
x60=65.9734457253857x_{60} = 65.9734457253857
x61=−34.5575191894877x_{61} = -34.5575191894877
x62=−72.2566310325652x_{62} = -72.2566310325652
x63=50.2256674407532x_{63} = 50.2256674407532
x64=−75.3716900810604x_{64} = -75.3716900810604
x65=−28.2743338823081x_{65} = -28.2743338823081
x66=−56.5132815466599x_{66} = -56.5132815466599
x67=−5.95017264337656x_{67} = -5.95017264337656
x68=53.4070751110265x_{68} = 53.4070751110265
x69=−47.1238898038469x_{69} = -47.1238898038469
Signos de extremos en los puntos:
(-50.22566744075319, -0.0398044981539202)

(-87.94185298068903, -0.022739359696793)

(25.0529526753384, 0.0797039218326035)

(84.82300164692441, 0)

(47.1238898038469, 0)

(100.51106726511297, 0.0198963368185454)

(-53.40707511102649, 0)

(-37.645997836015106, -0.0530890372838442)

(91.106186954104, 0)

(-84.82300164692441, 0)

(5.9501726433765585, 0.326891661078669)

(-3.141592653589793, 0)

(-69.08609101329898, -0.028943323105097)

(87.94185298068903, 0.022739359696793)

(-279.6017461694916, 0)

(-40.840704496667314, 0)

(62.80000863372525, 0.0318390562713079)

(-62.80000863372525, -0.0318390562713079)

(-100.51106726511297, -0.0198963368185454)

(-12.405499633586086, -0.160178002058028)

(78.53981633974483, 0)

(477.5178950270596, 0.00418830634377207)

(-18.742950211711907, -0.106403899511075)

(-9.42477796076938, 0)

(94.22655496545507, 0.0212230487092482)

(72.25663103256524, 0)

(56.51328154665989, 0.0353788334069361)

(9.42477796076938, 0)

(43.936785063740594, 0.0454963762334591)

(40.840704496667314, 0)

(59.69026041820607, 0)

(-25.0529526753384, -0.0797039218326035)

(-91.106186954104, 0)

(97.3893722612836, 0)

(34.55751918948773, 0)

(75.37169008106044, 0.026530491785468)

(-78.53981633974483, 0)

(37.645997836015106, 0.0530890372838442)

(31.352156690388735, 0.0637266332931457)

(779.1124110680094, 0.0025670194400556)

(81.6569174978428, 0.0244890470959608)

(-65.97344572538566, 0)

(-43.936785063740594, -0.0454963762334591)

(12.405499633586086, 0.160178002058028)

(3.141592653589793, 0)

(15.707963267948966, 0)

(-31.352156690388735, -0.0637266332931457)

(-21.991148575128552, 0)

(18.742950211711907, 0.106403899511075)

(-15.707963267948966, 0)

(69.08609101329898, 0.028943323105097)

(128.80529879718154, 0)

(-94.22655496545507, -0.0212230487092482)

(28.274333882308138, 0)

(-59.69026041820607, 0)

(-408.4021478425674, -0.00489710453208163)

(-81.6569174978428, -0.0244890470959608)

(-97.3893722612836, 0)

(21.991148575128552, 0)

(65.97344572538566, 0)

(-34.55751918948773, 0)

(-72.25663103256524, 0)

(50.22566744075319, 0.0398044981539202)

(-75.37169008106044, -0.026530491785468)

(-28.274333882308138, 0)

(-56.51328154665989, -0.0353788334069361)

(-5.9501726433765585, -0.326891661078669)

(53.40707511102649, 0)

(-47.1238898038469, 0)


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=−50.2256674407532x_{1} = -50.2256674407532
x2=−87.941852980689x_{2} = -87.941852980689
x3=84.8230016469244x_{3} = 84.8230016469244
x4=47.1238898038469x_{4} = 47.1238898038469
x5=−37.6459978360151x_{5} = -37.6459978360151
x6=91.106186954104x_{6} = 91.106186954104
x7=−69.086091013299x_{7} = -69.086091013299
x8=−62.8000086337252x_{8} = -62.8000086337252
x9=−100.511067265113x_{9} = -100.511067265113
x10=−12.4054996335861x_{10} = -12.4054996335861
x11=78.5398163397448x_{11} = 78.5398163397448
x12=−18.7429502117119x_{12} = -18.7429502117119
x13=72.2566310325652x_{13} = 72.2566310325652
x14=9.42477796076938x_{14} = 9.42477796076938
x15=40.8407044966673x_{15} = 40.8407044966673
x16=59.6902604182061x_{16} = 59.6902604182061
x17=−25.0529526753384x_{17} = -25.0529526753384
x18=97.3893722612836x_{18} = 97.3893722612836
x19=34.5575191894877x_{19} = 34.5575191894877
x20=−43.9367850637406x_{20} = -43.9367850637406
x21=3.14159265358979x_{21} = 3.14159265358979
x22=15.707963267949x_{22} = 15.707963267949
x23=−31.3521566903887x_{23} = -31.3521566903887
x24=128.805298797182x_{24} = 128.805298797182
x25=−94.2265549654551x_{25} = -94.2265549654551
x26=28.2743338823081x_{26} = 28.2743338823081
x27=−408.402147842567x_{27} = -408.402147842567
x28=−81.6569174978428x_{28} = -81.6569174978428
x29=21.9911485751286x_{29} = 21.9911485751286
x30=65.9734457253857x_{30} = 65.9734457253857
x31=−75.3716900810604x_{31} = -75.3716900810604
x32=−56.5132815466599x_{32} = -56.5132815466599
x33=−5.95017264337656x_{33} = -5.95017264337656
x34=53.4070751110265x_{34} = 53.4070751110265
Puntos máximos de la función:
x34=25.0529526753384x_{34} = 25.0529526753384
x34=100.511067265113x_{34} = 100.511067265113
x34=−53.4070751110265x_{34} = -53.4070751110265
x34=−84.8230016469244x_{34} = -84.8230016469244
x34=5.95017264337656x_{34} = 5.95017264337656
x34=−3.14159265358979x_{34} = -3.14159265358979
x34=87.941852980689x_{34} = 87.941852980689
x34=−279.601746169492x_{34} = -279.601746169492
x34=−40.8407044966673x_{34} = -40.8407044966673
x34=62.8000086337252x_{34} = 62.8000086337252
x34=477.51789502706x_{34} = 477.51789502706
x34=−9.42477796076938x_{34} = -9.42477796076938
x34=94.2265549654551x_{34} = 94.2265549654551
x34=56.5132815466599x_{34} = 56.5132815466599
x34=43.9367850637406x_{34} = 43.9367850637406
x34=−91.106186954104x_{34} = -91.106186954104
x34=75.3716900810604x_{34} = 75.3716900810604
x34=−78.5398163397448x_{34} = -78.5398163397448
x34=37.6459978360151x_{34} = 37.6459978360151
x34=31.3521566903887x_{34} = 31.3521566903887
x34=779.112411068009x_{34} = 779.112411068009
x34=81.6569174978428x_{34} = 81.6569174978428
x34=−65.9734457253857x_{34} = -65.9734457253857
x34=12.4054996335861x_{34} = 12.4054996335861
x34=−21.9911485751286x_{34} = -21.9911485751286
x34=18.7429502117119x_{34} = 18.7429502117119
x34=−15.707963267949x_{34} = -15.707963267949
x34=69.086091013299x_{34} = 69.086091013299
x34=−59.6902604182061x_{34} = -59.6902604182061
x34=−97.3893722612836x_{34} = -97.3893722612836
x34=−34.5575191894877x_{34} = -34.5575191894877
x34=−72.2566310325652x_{34} = -72.2566310325652
x34=50.2256674407532x_{34} = 50.2256674407532
x34=−28.2743338823081x_{34} = -28.2743338823081
x34=−47.1238898038469x_{34} = -47.1238898038469
Decrece en los intervalos
[128.805298797182,∞)\left[128.805298797182, \infty\right)
Crece en los intervalos
(−∞,−408.402147842567]\left(-\infty, -408.402147842567\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
−cos⁡(x)+2sin⁡(x)x+2(cos⁡(x)+1)x2x=0\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \left(\cos{\left(x \right)} + 1\right)}{x^{2}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=32.9240947361922x_{1} = 32.9240947361922
x2=23.4802919264971x_{2} = 23.4802919264971
x3=−98.9401572801928x_{3} = -98.9401572801928
x4=−83.2278838695937x_{4} = -83.2278838695937
x5=−64.3710918861937x_{5} = -64.3710918861937
x6=92.6556292628207x_{6} = 92.6556292628207
x7=39.2175881820314x_{7} = 39.2175881820314
x8=−42.3653892667121x_{8} = -42.3653892667121
x9=36.0743829885085x_{9} = 36.0743829885085
x10=−7.55102453615362x_{10} = -7.55102453615362
x11=86.3709080585407x_{11} = 86.3709080585407
x12=−80.0859487309835x_{12} = -80.0859487309835
x13=45.5081654611057x_{13} = 45.5081654611057
x14=−10.8268690995624x_{14} = -10.8268690995624
x15=4428.0743936732x_{15} = 4428.0743936732
x16=17.1687916817231x_{16} = 17.1687916817231
x17=95.7974791095636x_{17} = 95.7974791095636
x18=−17.1687916817231x_{18} = -17.1687916817231
x19=−92.6556292628207x_{19} = -92.6556292628207
x20=−86.3709080585407x_{20} = -86.3709080585407
x21=83.2278838695937x_{21} = 83.2278838695937
x22=64.3710918861937x_{22} = 64.3710918861937
x23=48.6544131811461x_{23} = 48.6544131811461
x24=61.2289201135729x_{24} = 61.2289201135729
x25=−32.9240947361922x_{25} = -32.9240947361922
x26=−36.0743829885085x_{26} = -36.0743829885085
x27=−89.5127959855887x_{27} = -89.5127959855887
x28=−70.6571246114187x_{28} = -70.6571246114187
x29=42.3653892667121x_{29} = 42.3653892667121
x30=−61.2289201135729x_{30} = -61.2289201135729
x31=−73.8006912312722x_{31} = -73.8006912312722
x32=80.0859487309835x_{32} = 80.0859487309835
x33=−13.9834279458844x_{33} = -13.9834279458844
x34=−23.4802919264971x_{34} = -23.4802919264971
x35=51.7969113967309x_{35} = 51.7969113967309
x36=−95.7974791095636x_{36} = -95.7974791095636
x37=7.55102453615362x_{37} = 7.55102453615362
x38=67.5150534592896x_{38} = 67.5150534592896
x39=58.0844318525335x_{39} = 58.0844318525335
x40=76.9426858848596x_{40} = 76.9426858848596
x41=−20.3169083532025x_{41} = -20.3169083532025
x42=−51.7969113967309x_{42} = -51.7969113967309
x43=26.6255299708041x_{43} = 26.6255299708041
x44=−39.2175881820314x_{44} = -39.2175881820314
x45=−76.9426858848596x_{45} = -76.9426858848596
x46=4.34230123285199x_{46} = 4.34230123285199
x47=−29.7801761695834x_{47} = -29.7801761695834
x48=13.9834279458844x_{48} = 13.9834279458844
x49=−48.6544131811461x_{49} = -48.6544131811461
x50=89.5127959855887x_{50} = 89.5127959855887
x51=−45.5081654611057x_{51} = -45.5081654611057
x52=70.6571246114187x_{52} = 70.6571246114187
x53=20.3169083532025x_{53} = 20.3169083532025
x54=54.9421240104386x_{54} = 54.9421240104386
x55=10.8268690995624x_{55} = 10.8268690995624
x56=29.7801761695834x_{56} = 29.7801761695834
x57=98.9401572801928x_{57} = 98.9401572801928
x58=73.8006912312722x_{58} = 73.8006912312722
x59=−67.5150534592896x_{59} = -67.5150534592896
x60=−26.6255299708041x_{60} = -26.6255299708041
x61=−54.9421240104386x_{61} = -54.9421240104386
x62=−4.34230123285199x_{62} = -4.34230123285199
x63=−58.0844318525335x_{63} = -58.0844318525335
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

lim⁡x→0−(−cos⁡(x)+2sin⁡(x)x+2(cos⁡(x)+1)x2x)=−∞\lim_{x \to 0^-}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \left(\cos{\left(x \right)} + 1\right)}{x^{2}}}{x}\right) = -\infty
lim⁡x→0+(−cos⁡(x)+2sin⁡(x)x+2(cos⁡(x)+1)x2x)=∞\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \left(\cos{\left(x \right)} + 1\right)}{x^{2}}}{x}\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- 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
[95.7974791095636,∞)\left[95.7974791095636, \infty\right)
Convexa en los intervalos
(−∞,−95.7974791095636]\left(-\infty, -95.7974791095636\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(cos⁡(x)+1x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} + 1}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(cos⁡(x)+1x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} + 1}{x}\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 (1 + cos(x))/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x)+1x2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} + 1}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x)+1x2)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} + 1}{x^{2}}\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:
cos⁡(x)+1x=−cos⁡(x)+1x\frac{\cos{\left(x \right)} + 1}{x} = - \frac{\cos{\left(x \right)} + 1}{x}
- No
cos⁡(x)+1x=cos⁡(x)+1x\frac{\cos{\left(x \right)} + 1}{x} = \frac{\cos{\left(x \right)} + 1}{x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (1+cos(x))/x
    © Sr Examen – Калькулятор