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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)=1−cos⁡(x)xf{\left(x \right)} = \frac{1 - \cos{\left(x \right)}}{x} 
f = (1 - cos(x))/x
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:
1−cos⁡(x)x=0\frac{1 - \cos{\left(x \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=2πx_{1} = 2 \pi
Solución numérica
x1=94.2477796093523x_{1} = 94.2477796093523
x2=−81.6814084756666x_{2} = -81.6814084756666
x3=−81.6814090381737x_{3} = -81.6814090381737
x4=−43.9822975863426x_{4} = -43.9822975863426
x5=31.4159260423149x_{5} = 31.4159260423149
x6=−94.2477794446532x_{6} = -94.2477794446532
x7=−43.9822971745099x_{7} = -43.9822971745099
x8=31.415926308189x_{8} = 31.415926308189
x9=−56.5486674648664x_{9} = -56.5486674648664
x10=−31.4159259860225x_{10} = -31.4159259860225
x11=−43.9822967286735x_{11} = -43.9822967286735
x12=18.8495570640723x_{12} = 18.8495570640723
x13=−87.9645938977678x_{13} = -87.9645938977678
x14=−62.8318534718064x_{14} = -62.8318534718064
x15=−6.28318511522188x_{15} = -6.28318511522188
x16=31.4159268388313x_{16} = 31.4159268388313
x17=87.9645937941679x_{17} = 87.9645937941679
x18=87.9645943358739x_{18} = 87.9645943358739
x19=−69.1150382719596x_{19} = -69.1150382719596
x20=414.69022329735x_{20} = 414.69022329735
x21=100.530965066794x_{21} = 100.530965066794
x22=37.6991120290429x_{22} = 37.6991120290429
x23=81.6814084733938x_{23} = 81.6814084733938
x24=75.3982232099398x_{24} = 75.3982232099398
x25=−37.6991113050299x_{25} = -37.6991113050299
x26=−257.610600184006x_{26} = -257.610600184006
x27=62.8318534171363x_{27} = 62.8318534171363
x28=−6.28318567459441x_{28} = -6.28318567459441
x29=94.2477792338511x_{29} = 94.2477792338511
x30=37.699111216894x_{30} = 37.699111216894
x31=87.964594500812x_{31} = 87.964594500812
x32=−37.6991118771954x_{32} = -37.6991118771954
x33=−94.2477801054162x_{33} = -94.2477801054162
x34=50.2654824463338x_{34} = 50.2654824463338
x35=18.8495556162397x_{35} = 18.8495556162397
x36=87.9645945664312x_{36} = 87.9645945664312
x37=6.2831855753381x_{37} = 6.2831855753381
x38=−43.9822979138367x_{38} = -43.9822979138367
x39=−25.1327415213295x_{39} = -25.1327415213295
x40=−6.28318586323633x_{40} = -6.28318586323633
x41=62.8318535453352x_{41} = 62.8318535453352
x42=−6.28318444067293x_{42} = -6.28318444067293
x43=6.2831852840235x_{43} = 6.2831852840235
x44=−69.1150378944947x_{44} = -69.1150378944947
x45=31.4159260114669x_{45} = 31.4159260114669
x46=12.566370437228x_{46} = 12.566370437228
x47=43.9822966284625x_{47} = 43.9822966284625
x48=−12.566371038017x_{48} = -12.566371038017
x49=12.5663702994139x_{49} = 12.5663702994139
x50=−87.9645947491978x_{50} = -87.9645947491978
x51=−69.1150386836456x_{51} = -69.1150386836456
x52=56.5486676001308x_{52} = 56.5486676001308
x53=69.1150379829339x_{53} = 69.1150379829339
x54=−81.6814092106571x_{54} = -81.6814092106571
x55=100.530964759279x_{55} = 100.530964759279
x56=43.9822971694463x_{56} = 43.9822971694463
x57=62.8318527817653x_{57} = 62.8318527817653
x58=−12.5663710453539x_{58} = -12.5663710453539
x59=−100.53096462403x_{59} = -100.53096462403
x60=50.265482070609x_{60} = 50.265482070609
x61=138.230076849674x_{61} = 138.230076849674
x62=−37.6991120342542x_{62} = -37.6991120342542
x63=−31.4159256195136x_{63} = -31.4159256195136
x64=56.5486679086989x_{64} = 56.5486679086989
x65=−75.3982238731219x_{65} = -75.3982238731219
x66=81.6814091886423x_{66} = 81.6814091886423
x67=−75.3982231582427x_{67} = -75.3982231582427
x68=25.1327408158131x_{68} = 25.1327408158131
x69=6.28318444287659x_{69} = 6.28318444287659
x70=−94.2477796479604x_{70} = -94.2477796479604
x71=−1237.7874595289x_{71} = -1237.7874595289
x72=−75.3982229197268x_{72} = -75.3982229197268
x73=25.1327416214209x_{73} = 25.1327416214209
x74=43.9822974012399x_{74} = 43.9822974012399
x75=75.3982239999848x_{75} = 75.3982239999848
x76=81.6814084412002x_{76} = 81.6814084412002
x77=50.2654829117362x_{77} = 50.2654829117362
x78=94.2477800713522x_{78} = 94.2477800713522
x79=12.5663710535352x_{79} = 12.5663710535352
x80=−50.2654824779719x_{80} = -50.2654824779719
x81=−31.4159267135067x_{81} = -31.4159267135067
x82=−56.5486685316992x_{82} = -56.5486685316992
x83=−56.5486682301784x_{83} = -56.5486682301784
x84=−18.8495563014918x_{84} = -18.8495563014918
x85=18.8495563674798x_{85} = 18.8495563674798
x86=−100.530965615463x_{86} = -100.530965615463
x87=−50.2654829455153x_{87} = -50.2654829455153
x88=−12.5663702928087x_{88} = -12.5663702928087
x89=56.5486682620429x_{89} = 56.5486682620429
x90=69.1150387875051x_{90} = 69.1150387875051
x91=−87.9645943585322x_{91} = -87.9645943585322
x92=−25.1327407213372x_{92} = -25.1327407213372
x93=−18.8495554933496x_{93} = -18.8495554933496
x94=−50.2654822850339x_{94} = -50.2654822850339
x95=37.6991113062675x_{95} = 37.6991113062675
x96=−62.8318526665123x_{96} = -62.8318526665123
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)=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
sin⁡(x)x−1−cos⁡(x)x2=0\frac{\sin{\left(x \right)}}{x} - \frac{1 - \cos{\left(x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−62.8318530717959x_{1} = -62.8318530717959
x2=87.9645943005142x_{2} = 87.9645943005142
x3=−78.5143446648172x_{3} = -78.5143446648172
x4=31.4159265358979x_{4} = 31.4159265358979
x5=−56.5486677646163x_{5} = -56.5486677646163
x6=69.1150383789755x_{6} = 69.1150383789755
x7=−21.8998872970823x_{7} = -21.8998872970823
x8=−37.6991118430775x_{8} = -37.6991118430775
x9=−81.6814089933346x_{9} = -81.6814089933346
x10=169.646003293849x_{10} = 169.646003293849
x11=−15.5797675022891x_{11} = -15.5797675022891
x12=−59.656738426191x_{12} = -59.656738426191
x13=−12.5663706143592x_{13} = -12.5663706143592
x14=−97.3688325296866x_{14} = -97.3688325296866
x15=−87.9645943005142x_{15} = -87.9645943005142
x16=12.5663706143592x_{16} = 12.5663706143592
x17=−100.530964914873x_{17} = -100.530964914873
x18=−1181.23883774976x_{18} = -1181.23883774976
x19=−2.33112237041442x_{19} = -2.33112237041442
x20=−40.7916847146183x_{20} = -40.7916847146183
x21=78.5143446648172x_{21} = 78.5143446648172
x22=40.7916847146183x_{22} = 40.7916847146183
x23=−47.0814165846103x_{23} = -47.0814165846103
x24=−84.7994176724893x_{24} = -84.7994176724893
x25=−94.2477796076938x_{25} = -94.2477796076938
x26=6.28318530717959x_{26} = 6.28318530717959
x27=−65.943118880897x_{27} = -65.943118880897
x28=−69.1150383789755x_{28} = -69.1150383789755
x29=−53.3696049818501x_{29} = -53.3696049818501
x30=21.8998872970823x_{30} = 21.8998872970823
x31=−50.2654824574367x_{31} = -50.2654824574367
x32=−25.1327412287183x_{32} = -25.1327412287183
x33=18.8495559215388x_{33} = 18.8495559215388
x34=−18.8495559215388x_{34} = -18.8495559215388
x35=59.656738426191x_{35} = 59.656738426191
x36=37.6991118430775x_{36} = 37.6991118430775
x37=−43.9822971502571x_{37} = -43.9822971502571
x38=−6.28318530717959x_{38} = -6.28318530717959
x39=−9.20843355440115x_{39} = -9.20843355440115
x40=65.943118880897x_{40} = 65.943118880897
x41=91.0842301384618x_{41} = 91.0842301384618
x42=92394.2399420758x_{42} = 92394.2399420758
x43=43.9822971502571x_{43} = 43.9822971502571
x44=56.5486677646163x_{44} = 56.5486677646163
x45=34.4995636692158x_{45} = 34.4995636692158
x46=47.0814165846103x_{46} = 47.0814165846103
x47=25.1327412287183x_{47} = 25.1327412287183
x48=28.2034502671317x_{48} = 28.2034502671317
x49=75.398223686155x_{49} = 75.398223686155
x50=−91.0842301384618x_{50} = -91.0842301384618
x51=81.6814089933346x_{51} = 81.6814089933346
x52=2.33112237041442x_{52} = 2.33112237041442
x53=−28.2034502671317x_{53} = -28.2034502671317
x54=100.530964914873x_{54} = 100.530964914873
x55=72.2289430706097x_{55} = 72.2289430706097
x56=−34.4995636692158x_{56} = -34.4995636692158
x57=−75.398223686155x_{57} = -75.398223686155
x58=−31.4159265358979x_{58} = -31.4159265358979
x59=9.20843355440115x_{59} = 9.20843355440115
x60=−103.653263067797x_{60} = -103.653263067797
x61=−72.2289430706097x_{61} = -72.2289430706097
x62=84.7994176724893x_{62} = 84.7994176724893
x63=15.5797675022891x_{63} = 15.5797675022891
x64=62.8318530717959x_{64} = 62.8318530717959
x65=−307.8760800518x_{65} = -307.8760800518
x66=50.2654824574367x_{66} = 50.2654824574367
x67=53.3696049818501x_{67} = 53.3696049818501
x68=94.2477796076938x_{68} = 94.2477796076938
x69=97.3688325296866x_{69} = 97.3688325296866
Signos de extremos en los puntos:
(-62.83185307179586, 0)

(87.96459430051421, 0)

(-78.51434466481717, -0.0254689206534694)

(31.41592653589793, 0)

(-56.548667764616276, 0)

(69.11503837897546, 0)

(-21.89988729708232, -0.0911346506917966)

(-37.69911184307752, 0)

(-81.68140899333463, 0)

(169.64600329384882, 0)

(-15.579767502289146, -0.127844922574794)

(-59.65673842619101, -0.0335157141235985)

(-12.566370614359172, 0)

(-97.36883252968656, -0.0205382874085413)

(-87.96459430051421, 0)

(12.566370614359172, 0)

(-100.53096491487338, 0)

(-1181.2388377497623, 0)

(-2.331122370414423, -0.724611353776708)

(-40.791684714618334, -0.0490001524829528)

(78.51434466481717, 0.0254689206534694)

(40.791684714618334, 0.0490001524829528)

(-47.0814165846103, -0.0424604502887016)

(-84.79941767248933, -0.0235817882463307)

(-94.2477796076938, 0)

(6.283185307179586, 0)

(-65.94311888089696, -0.0303221960142206)

(-69.11503837897546, 0)

(-53.36960498185014, -0.0374613617155508)

(21.89988729708232, 0.0911346506917966)

(-50.26548245743669, 0)

(-25.132741228718345, 0)

(18.84955592153876, 0)

(-18.84955592153876, 0)

(59.65673842619101, 0.0335157141235985)

(37.69911184307752, 0)

(-43.982297150257104, 0)

(-6.283185307179586, 0)

(-9.208433554401154, -0.214660688386019)

(65.94311888089696, 0.0303221960142206)

(91.0842301384618, 0.021955051448177)

(92394.23994207582, 0)

(43.982297150257104, 0)

(56.548667764616276, 0)

(34.49956366921579, 0.0579230818110724)

(47.0814165846103, 0.0424604502887016)

(25.132741228718345, 0)

(28.203450267131746, 0.0708242711210408)

(75.39822368615503, 0)

(-91.0842301384618, -0.021955051448177)

(81.68140899333463, 0)

(2.331122370414423, 0.724611353776708)

(-28.203450267131746, -0.0708242711210408)

(100.53096491487338, 0)

(72.2289430706097, 0.0276844243853039)

(-34.49956366921579, -0.0579230818110724)

(-75.39822368615503, 0)

(-31.41592653589793, 0)

(9.208433554401154, 0.214660688386019)

(-103.65326306779691, -0.0192933035363155)

(-72.2289430706097, -0.0276844243853039)

(84.79941767248933, 0.0235817882463307)

(15.579767502289146, 0.127844922574794)

(62.83185307179586, 0)

(-307.87608005179976, 0)

(50.26548245743669, 0)

(53.36960498185014, 0.0374613617155508)

(94.2477796076938, 0)

(97.36883252968656, 0.0205382874085413)


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=87.9645943005142x_{1} = 87.9645943005142
x2=−78.5143446648172x_{2} = -78.5143446648172
x3=31.4159265358979x_{3} = 31.4159265358979
x4=69.1150383789755x_{4} = 69.1150383789755
x5=−21.8998872970823x_{5} = -21.8998872970823
x6=169.646003293849x_{6} = 169.646003293849
x7=−15.5797675022891x_{7} = -15.5797675022891
x8=−59.656738426191x_{8} = -59.656738426191
x9=−97.3688325296866x_{9} = -97.3688325296866
x10=12.5663706143592x_{10} = 12.5663706143592
x11=−2.33112237041442x_{11} = -2.33112237041442
x12=−40.7916847146183x_{12} = -40.7916847146183
x13=−47.0814165846103x_{13} = -47.0814165846103
x14=−84.7994176724893x_{14} = -84.7994176724893
x15=6.28318530717959x_{15} = 6.28318530717959
x16=−65.943118880897x_{16} = -65.943118880897
x17=−53.3696049818501x_{17} = -53.3696049818501
x18=18.8495559215388x_{18} = 18.8495559215388
x19=37.6991118430775x_{19} = 37.6991118430775
x20=−9.20843355440115x_{20} = -9.20843355440115
x21=92394.2399420758x_{21} = 92394.2399420758
x22=43.9822971502571x_{22} = 43.9822971502571
x23=56.5486677646163x_{23} = 56.5486677646163
x24=25.1327412287183x_{24} = 25.1327412287183
x25=75.398223686155x_{25} = 75.398223686155
x26=−91.0842301384618x_{26} = -91.0842301384618
x27=81.6814089933346x_{27} = 81.6814089933346
x28=−28.2034502671317x_{28} = -28.2034502671317
x29=100.530964914873x_{29} = 100.530964914873
x30=−34.4995636692158x_{30} = -34.4995636692158
x31=−103.653263067797x_{31} = -103.653263067797
x32=−72.2289430706097x_{32} = -72.2289430706097
x33=62.8318530717959x_{33} = 62.8318530717959
x34=50.2654824574367x_{34} = 50.2654824574367
x35=94.2477796076938x_{35} = 94.2477796076938
Puntos máximos de la función:
x35=−62.8318530717959x_{35} = -62.8318530717959
x35=−56.5486677646163x_{35} = -56.5486677646163
x35=−37.6991118430775x_{35} = -37.6991118430775
x35=−81.6814089933346x_{35} = -81.6814089933346
x35=−12.5663706143592x_{35} = -12.5663706143592
x35=−87.9645943005142x_{35} = -87.9645943005142
x35=−100.530964914873x_{35} = -100.530964914873
x35=−1181.23883774976x_{35} = -1181.23883774976
x35=78.5143446648172x_{35} = 78.5143446648172
x35=40.7916847146183x_{35} = 40.7916847146183
x35=−94.2477796076938x_{35} = -94.2477796076938
x35=−69.1150383789755x_{35} = -69.1150383789755
x35=21.8998872970823x_{35} = 21.8998872970823
x35=−50.2654824574367x_{35} = -50.2654824574367
x35=−25.1327412287183x_{35} = -25.1327412287183
x35=−18.8495559215388x_{35} = -18.8495559215388
x35=59.656738426191x_{35} = 59.656738426191
x35=−43.9822971502571x_{35} = -43.9822971502571
x35=−6.28318530717959x_{35} = -6.28318530717959
x35=65.943118880897x_{35} = 65.943118880897
x35=91.0842301384618x_{35} = 91.0842301384618
x35=34.4995636692158x_{35} = 34.4995636692158
x35=47.0814165846103x_{35} = 47.0814165846103
x35=28.2034502671317x_{35} = 28.2034502671317
x35=2.33112237041442x_{35} = 2.33112237041442
x35=72.2289430706097x_{35} = 72.2289430706097
x35=−75.398223686155x_{35} = -75.398223686155
x35=−31.4159265358979x_{35} = -31.4159265358979
x35=9.20843355440115x_{35} = 9.20843355440115
x35=84.7994176724893x_{35} = 84.7994176724893
x35=15.5797675022891x_{35} = 15.5797675022891
x35=−307.8760800518x_{35} = -307.8760800518
x35=53.3696049818501x_{35} = 53.3696049818501
x35=97.3688325296866x_{35} = 97.3688325296866
Decrece en los intervalos
[92394.2399420758,∞)\left[92394.2399420758, \infty\right)
Crece en los intervalos
(−∞,−103.653263067797]\left(-\infty, -103.653263067797\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=−61.2278525685536x_{1} = -61.2278525685536
x2=−17.1551175692195x_{2} = -17.1551175692195
x3=14.0040665914265x_{3} = 14.0040665914265
x4=7.62307729555873x_{4} = 7.62307729555873
x5=83.2284614928992x_{5} = 83.2284614928992
x6=70.6579261366639x_{6} = 70.6579261366639
x7=−67.5141755390553x_{7} = -67.5141755390553
x8=−26.6311871536774x_{8} = -26.6311871536774
x9=32.927791200115x_{9} = 32.927791200115
x10=−39.220192145926x_{10} = -39.220192145926
x11=−76.943361762789x_{11} = -76.943361762789
x12=−83.2284614928992x_{12} = -83.2284614928992
x13=−23.4730079186956x_{13} = -23.4730079186956
x14=17.1551175692195x_{14} = 17.1551175692195
x15=80.0853248723334x_{15} = 80.0853248723334
x16=92.6551632265069x_{16} = 92.6551632265069
x17=−73.7999565431799x_{17} = -73.7999565431799
x18=61.2278525685536x_{18} = 61.2278525685536
x19=−36.0713042845073x_{19} = -36.0713042845073
x20=39.220192145926x_{20} = 39.220192145926
x21=36.0713042845073x_{21} = 36.0713042845073
x22=−7.62307729555873x_{22} = -7.62307729555873
x23=76.943361762789x_{23} = 76.943361762789
x24=−51.7984033793852x_{24} = -51.7984033793852
x25=−54.9407979981311x_{25} = -54.9407979981311
x26=168.063235595213x_{26} = 168.063235595213
x27=−4.08557388547682x_{27} = -4.08557388547682
x28=755.55038961184x_{28} = 755.55038961184
x29=−89.5132953248997x_{29} = -89.5132953248997
x30=67.5141755390553x_{30} = 67.5141755390553
x31=−86.3703717136003x_{31} = -86.3703717136003
x32=−20.3266414701876x_{32} = -20.3266414701876
x33=20.3266414701876x_{33} = 20.3266414701876
x34=64.3720576734236x_{34} = 64.3720576734236
x35=−98.9397485795284x_{35} = -98.9397485795284
x36=23.4730079186956x_{36} = 23.4730079186956
x37=−58.0856181381215x_{37} = -58.0856181381215
x38=−92.6551632265069x_{38} = -92.6551632265069
x39=−80.0853248723334x_{39} = -80.0853248723334
x40=−29.7756549714707x_{40} = -29.7756549714707
x41=−48.6527219697238x_{41} = -48.6527219697238
x42=26.6311871536774x_{42} = 26.6311871536774
x43=29.7756549714707x_{43} = 29.7756549714707
x44=−70.6579261366639x_{44} = -70.6579261366639
x45=42.3631580330254x_{45} = 42.3631580330254
x46=58.0856181381215x_{46} = 58.0856181381215
x47=45.5100986876418x_{47} = 45.5100986876418
x48=−180.630443726271x_{48} = -180.630443726271
x49=98.9397485795284x_{49} = 98.9397485795284
x50=−14.0040665914265x_{50} = -14.0040665914265
x51=73.7999565431799x_{51} = 73.7999565431799
x52=48.6527219697238x_{52} = 48.6527219697238
x53=10.7920322357124x_{53} = 10.7920322357124
x54=−10.7920322357124x_{54} = -10.7920322357124
x55=−32.927791200115x_{55} = -32.927791200115
x56=−42.3631580330254x_{56} = -42.3631580330254
x57=86.3703717136003x_{57} = 86.3703717136003
x58=−485.371952906199x_{58} = -485.371952906199
x59=−95.7979150674353x_{59} = -95.7979150674353
x60=95.7979150674353x_{60} = 95.7979150674353
x61=54.9407979981311x_{61} = 54.9407979981311
x62=51.7984033793852x_{62} = 51.7984033793852
x63=89.5132953248997x_{63} = 89.5132953248997
x64=−45.5100986876418x_{64} = -45.5100986876418
x65=−64.3720576734236x_{65} = -64.3720576734236
x66=4.08557388547682x_{66} = 4.08557388547682
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)=0\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) = 0
lim⁡x→0+(cos⁡(x)−2sin⁡(x)x−2(cos⁡(x)−1)x2x)=0\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) = 0
- los límites son iguales, es decir omitimos el punto correspondiente

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
[168.063235595213,∞)\left[168.063235595213, \infty\right)
Convexa en los intervalos
(−∞,−180.630443726271]\left(-\infty, -180.630443726271\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→−∞(1−cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(1−cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{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→−∞(1−cos⁡(x)x2)=0\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(1−cos⁡(x)x2)=0\lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{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:
1−cos⁡(x)x=−1−cos⁡(x)x\frac{1 - \cos{\left(x \right)}}{x} = - \frac{1 - \cos{\left(x \right)}}{x}
- No
1−cos⁡(x)x=1−cos⁡(x)x\frac{1 - \cos{\left(x \right)}}{x} = \frac{1 - \cos{\left(x \right)}}{x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (1-cos(x))/x
    © Sr Examen – Калькулятор