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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x - 1)
f(x) = ----------
         x - 1
f(x)=cos⁡(x−1)x−1f{\left(x \right)} = \frac{\cos{\left(x - 1 \right)}}{x - 1} 
f = cos(x - 1)/(x - 1)
Gráfico de la función
02468-8-6-4-2-1010-5050
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=1x_{1} = 1
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−1)x−1=0\frac{\cos{\left(x - 1 \right)}}{x - 1} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1+π2x_{1} = 1 + \frac{\pi}{2}
x2=1+3π2x_{2} = 1 + \frac{3 \pi}{2}
Solución numérica
x1=−91.6769832808989x_{1} = -91.6769832808989
x2=−82.2522053201295x_{2} = -82.2522053201295
x3=87.3937979737193x_{3} = 87.3937979737193
x4=90.5353906273091x_{4} = 90.5353906273091
x5=24.5619449019235x_{5} = 24.5619449019235
x6=84.2522053201295x_{6} = 84.2522053201295
x7=−9.99557428756428x_{7} = -9.99557428756428
x8=−75.9690200129499x_{8} = -75.9690200129499
x9=49.6946861306418x_{9} = 49.6946861306418
x10=−38.2699081698724x_{10} = -38.2699081698724
x11=30.845130209103x_{11} = 30.845130209103
x12=−94.8185759344887x_{12} = -94.8185759344887
x13=59.1194640914112x_{13} = 59.1194640914112
x14=55.9778714378214x_{14} = 55.9778714378214
x15=99.9601685880785x_{15} = 99.9601685880785
x16=21.4203522483337x_{16} = 21.4203522483337
x17=−101.101761241668x_{17} = -101.101761241668
x18=81.1106126665397x_{18} = 81.1106126665397
x19=−50.8362787842316x_{19} = -50.8362787842316
x20=−85.3937979737193x_{20} = -85.3937979737193
x21=−79.1106126665397x_{21} = -79.1106126665397
x22=−69.6858347057703x_{22} = -69.6858347057703
x23=71.6858347057703x_{23} = 71.6858347057703
x24=−13.1371669411541x_{24} = -13.1371669411541
x25=8.85398163397448x_{25} = 8.85398163397448
x26=−31.9867228626928x_{26} = -31.9867228626928
x27=−25.7035375555132x_{27} = -25.7035375555132
x28=−28.845130209103x_{28} = -28.845130209103
x29=−16.2787595947439x_{29} = -16.2787595947439
x30=62.261056745001x_{30} = 62.261056745001
x31=18.2787595947439x_{31} = 18.2787595947439
x32=−72.8274273593601x_{32} = -72.8274273593601
x33=−97.9601685880785x_{33} = -97.9601685880785
x34=27.7035375555132x_{34} = 27.7035375555132
x35=−0.570796326794897x_{35} = -0.570796326794897
x36=−6.85398163397448x_{36} = -6.85398163397448
x37=2.5707963267949x_{37} = 2.5707963267949
x38=−44.553093477052x_{38} = -44.553093477052
x39=−63.4026493985908x_{39} = -63.4026493985908
x40=68.5442420521806x_{40} = 68.5442420521806
x41=77.9690200129499x_{41} = 77.9690200129499
x42=65.4026493985908x_{42} = 65.4026493985908
x43=−214.199096770901x_{43} = -214.199096770901
x44=−3.71238898038469x_{44} = -3.71238898038469
x45=357.570766182442x_{45} = 357.570766182442
x46=125.092909816797x_{46} = 125.092909816797
x47=40.2699081698724x_{47} = 40.2699081698724
x48=103.101761241668x_{48} = 103.101761241668
x49=5.71238898038469x_{49} = 5.71238898038469
x50=−53.9778714378214x_{50} = -53.9778714378214
x51=15.1371669411541x_{51} = 15.1371669411541
x52=134.517687777566x_{52} = 134.517687777566
x53=52.8362787842316x_{53} = 52.8362787842316
x54=43.4115008234622x_{54} = 43.4115008234622
x55=109.384946548848x_{55} = 109.384946548848
x56=11.9955742875643x_{56} = 11.9955742875643
x57=46.553093477052x_{57} = 46.553093477052
x58=37.1283155162826x_{58} = 37.1283155162826
x59=74.8274273593601x_{59} = 74.8274273593601
x60=−57.1194640914112x_{60} = -57.1194640914112
x61=96.8185759344887x_{61} = 96.8185759344887
x62=−35.1283155162826x_{62} = -35.1283155162826
x63=−88.5353906273091x_{63} = -88.5353906273091
x64=−22.5619449019235x_{64} = -22.5619449019235
x65=−47.6946861306418x_{65} = -47.6946861306418
x66=−19.4203522483337x_{66} = -19.4203522483337
x67=−60.261056745001x_{67} = -60.261056745001
x68=−41.4115008234622x_{68} = -41.4115008234622
x69=33.9867228626928x_{69} = 33.9867228626928
x70=−66.5442420521806x_{70} = -66.5442420521806
x71=93.6769832808989x_{71} = 93.6769832808989
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 cos(x - 1)/(x - 1).
cos⁡(−1)−1\frac{\cos{\left(-1 \right)}}{-1}
Resultado:
f(0)=−cos⁡(1)f{\left(0 \right)} = - \cos{\left(1 \right)}
Punto:
(0, -cos(1))
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−1)x−1−cos⁡(x−1)(x−1)2=0- \frac{\sin{\left(x - 1 \right)}}{x - 1} - \frac{\cos{\left(x - 1 \right)}}{\left(x - 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−130.939312353727x_{1} = -130.939312353727
x2=−55.5309801938186x_{2} = -55.5309801938186
x3=44.9595528888955x_{3} = 44.9595528888955
x4=48.1026627703624x_{4} = 48.1026627703624
x5=−58.6735041304405x_{5} = -58.6735041304405
x6=−8.31786646179107x_{6} = -8.31786646179107
x7=98.3791034786112x_{7} = 98.3791034786112
x8=29.2389365752603x_{8} = 29.2389365752603
x9=−74.3849592185347x_{9} = -74.3849592185347
x10=79.5270825679419x_{10} = 79.5270825679419
x11=63.8159348889734x_{11} = 63.8159348889734
x12=−52.3883466217256x_{12} = -52.3883466217256
x13=10.3178664617911x_{13} = 10.3178664617911
x14=60.6735041304405x_{14} = 60.6735041304405
x15=−30.3840740178899x_{15} = -30.3840740178899
x16=70.100567727981x_{16} = 70.100567727981
x17=−17.7964043662102x_{17} = -17.7964043662102
x18=−90.0952098694071x_{18} = -90.0952098694071
x19=16.644128370333x_{19} = 16.644128370333
x20=−36.672573565113x_{20} = -36.672573565113
x21=57.5309801938186x_{21} = 57.5309801938186
x22=−96.3791034786112x_{22} = -96.3791034786112
x23=−5.12125046689807x_{23} = -5.12125046689807
x24=13.4864543952238x_{24} = 13.4864543952238
x25=−1.79838604578389x_{25} = -1.79838604578389
x26=−68.100567727981x_{26} = -68.100567727981
x27=334.005818339011x_{27} = 334.005818339011
x28=3.79838604578389x_{28} = 3.79838604578389
x29=−42.9595528888955x_{29} = -42.9595528888955
x30=82.6691650818489x_{30} = 82.6691650818489
x31=−99.5210170746866x_{31} = -99.5210170746866
x32=88.9532251106725x_{32} = 88.9532251106725
x33=92.0952098694071x_{33} = 92.0952098694071
x34=−49.2455828375744x_{34} = -49.2455828375744
x35=66.9582857893902x_{35} = 66.9582857893902
x36=−71.2427897046973x_{36} = -71.2427897046973
x37=41.8162093266346x_{37} = 41.8162093266346
x38=−14.644128370333x_{38} = -14.644128370333
x39=54.3883466217256x_{39} = 54.3883466217256
x40=85.811211299318x_{40} = 85.811211299318
x41=−93.2371684817036x_{41} = -93.2371684817036
x42=73.2427897046973x_{42} = 73.2427897046973
x43=−24.0929104121121x_{43} = -24.0929104121121
x44=35.5285657554621x_{44} = 35.5285657554621
x45=−80.6691650818489x_{45} = -80.6691650818489
x46=19.7964043662102x_{46} = 19.7964043662102
x47=−86.9532251106725x_{47} = -86.9532251106725
x48=−20.945612879981x_{48} = -20.945612879981
x49=−27.2389365752603x_{49} = -27.2389365752603
x50=32.3840740178899x_{50} = 32.3840740178899
x51=22.945612879981x_{51} = 22.945612879981
x52=51.2455828375744x_{52} = 51.2455828375744
x53=−83.811211299318x_{53} = -83.811211299318
x54=95.2371684817036x_{54} = 95.2371684817036
x55=−11.4864543952238x_{55} = -11.4864543952238
x56=76.3849592185347x_{56} = 76.3849592185347
x57=−61.8159348889734x_{57} = -61.8159348889734
x58=7.12125046689807x_{58} = 7.12125046689807
x59=−77.5270825679419x_{59} = -77.5270825679419
x60=−64.9582857893902x_{60} = -64.9582857893902
x61=−33.5285657554621x_{61} = -33.5285657554621
x62=26.0929104121121x_{62} = 26.0929104121121
x63=−39.8162093266346x_{63} = -39.8162093266346
x64=38.672573565113x_{64} = 38.672573565113
x65=−46.1026627703624x_{65} = -46.1026627703624
Signos de extremos en los puntos:
(-130.9393123537265, -0.00757902448438246)

(-55.53098019381864, -0.0176866485521696)

(44.959552888895495, 0.0227423004725314)

(48.10266277036235, -0.0212254394164143)

(-58.67350413044053, 0.0167555036571887)

(-8.317866461791066, 0.106707947715237)

(98.3791034786112, -0.0102686022030809)

(29.238936575260272, -0.0353899155541688)

(-74.38495921853475, -0.0132640786518247)

(79.52708256794193, -0.0127334276777468)

(63.81593488897342, 0.015917510583426)

(-52.38834662172563, 0.0187273944640866)

(10.317866461791066, -0.106707947715237)

(60.67350413044053, -0.0167555036571887)

(-30.38407401788986, -0.0318471321112693)

(70.10056772798097, 0.0144701459746764)

(-17.796404366210158, -0.0531265325613881)

(-90.09520986940714, 0.0109768642483425)

(16.64412837033303, -0.0637915530395936)

(-36.67257356511297, -0.0265351630103045)

(57.53098019381864, 0.0176866485521696)

(-96.3791034786112, 0.0102686022030809)

(-5.1212504668980685, -0.161228034325064)

(13.486454395223781, 0.0798311807800032)

(-1.7983860457838872, 0.336508416918395)

(-68.10056772798097, -0.0144701459746764)

(334.00581833901066, 0.00300293699420144)

(3.798386045783887, -0.336508416918395)

(-42.959552888895495, -0.0227423004725314)

(82.66916508184887, 0.0122436055670467)

(-99.52101707468658, -0.00994767611536293)

(88.95322511067255, 0.0113689449158811)

(92.09520986940714, -0.0109768642483425)

(-49.24558283757444, -0.0198983065303553)

(66.95828578939016, -0.0151593553168405)

(-71.24278970469729, 0.0138408859131547)

(41.81620932663458, -0.0244927205346957)

(-14.644128370333028, 0.0637915530395936)

(54.38834662172563, -0.0187273944640866)

(85.81121129931802, -0.0117900744410766)

(-93.23716848170359, -0.01061092686295)

(73.24278970469729, -0.0138408859131547)

(-24.092910412112097, -0.0398202855500511)

(35.52856575546206, -0.0289493889114503)

(-80.66916508184887, -0.0122436055670467)

(19.796404366210158, 0.0531265325613881)

(-86.95322511067255, -0.0113689449158811)

(-20.945612879981045, 0.0455199604051285)

(-27.238936575260272, 0.0353899155541688)

(32.38407401788986, 0.0318471321112693)

(22.945612879981045, -0.0455199604051285)

(51.24558283757444, 0.0198983065303553)

(-83.81121129931802, 0.0117900744410766)

(95.23716848170359, 0.01061092686295)

(-11.486454395223781, -0.0798311807800032)

(76.38495921853475, 0.0132640786518247)

(-61.81593488897342, -0.015917510583426)

(7.1212504668980685, 0.161228034325064)

(-77.52708256794193, 0.0127334276777468)

(-64.95828578939016, 0.0151593553168405)

(-33.52856575546206, 0.0289493889114503)

(26.092910412112097, 0.0398202855500511)

(-39.81620932663458, 0.0244927205346957)

(38.67257356511297, 0.0265351630103045)

(-46.10266277036235, 0.0212254394164143)


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=−130.939312353727x_{1} = -130.939312353727
x2=−55.5309801938186x_{2} = -55.5309801938186
x3=48.1026627703624x_{3} = 48.1026627703624
x4=98.3791034786112x_{4} = 98.3791034786112
x5=29.2389365752603x_{5} = 29.2389365752603
x6=−74.3849592185347x_{6} = -74.3849592185347
x7=79.5270825679419x_{7} = 79.5270825679419
x8=10.3178664617911x_{8} = 10.3178664617911
x9=60.6735041304405x_{9} = 60.6735041304405
x10=−30.3840740178899x_{10} = -30.3840740178899
x11=−17.7964043662102x_{11} = -17.7964043662102
x12=16.644128370333x_{12} = 16.644128370333
x13=−36.672573565113x_{13} = -36.672573565113
x14=−5.12125046689807x_{14} = -5.12125046689807
x15=−68.100567727981x_{15} = -68.100567727981
x16=3.79838604578389x_{16} = 3.79838604578389
x17=−42.9595528888955x_{17} = -42.9595528888955
x18=−99.5210170746866x_{18} = -99.5210170746866
x19=92.0952098694071x_{19} = 92.0952098694071
x20=−49.2455828375744x_{20} = -49.2455828375744
x21=66.9582857893902x_{21} = 66.9582857893902
x22=41.8162093266346x_{22} = 41.8162093266346
x23=54.3883466217256x_{23} = 54.3883466217256
x24=85.811211299318x_{24} = 85.811211299318
x25=−93.2371684817036x_{25} = -93.2371684817036
x26=73.2427897046973x_{26} = 73.2427897046973
x27=−24.0929104121121x_{27} = -24.0929104121121
x28=35.5285657554621x_{28} = 35.5285657554621
x29=−80.6691650818489x_{29} = -80.6691650818489
x30=−86.9532251106725x_{30} = -86.9532251106725
x31=22.945612879981x_{31} = 22.945612879981
x32=−11.4864543952238x_{32} = -11.4864543952238
x33=−61.8159348889734x_{33} = -61.8159348889734
Puntos máximos de la función:
x33=44.9595528888955x_{33} = 44.9595528888955
x33=−58.6735041304405x_{33} = -58.6735041304405
x33=−8.31786646179107x_{33} = -8.31786646179107
x33=63.8159348889734x_{33} = 63.8159348889734
x33=−52.3883466217256x_{33} = -52.3883466217256
x33=70.100567727981x_{33} = 70.100567727981
x33=−90.0952098694071x_{33} = -90.0952098694071
x33=57.5309801938186x_{33} = 57.5309801938186
x33=−96.3791034786112x_{33} = -96.3791034786112
x33=13.4864543952238x_{33} = 13.4864543952238
x33=−1.79838604578389x_{33} = -1.79838604578389
x33=334.005818339011x_{33} = 334.005818339011
x33=82.6691650818489x_{33} = 82.6691650818489
x33=88.9532251106725x_{33} = 88.9532251106725
x33=−71.2427897046973x_{33} = -71.2427897046973
x33=−14.644128370333x_{33} = -14.644128370333
x33=19.7964043662102x_{33} = 19.7964043662102
x33=−20.945612879981x_{33} = -20.945612879981
x33=−27.2389365752603x_{33} = -27.2389365752603
x33=32.3840740178899x_{33} = 32.3840740178899
x33=51.2455828375744x_{33} = 51.2455828375744
x33=−83.811211299318x_{33} = -83.811211299318
x33=95.2371684817036x_{33} = 95.2371684817036
x33=76.3849592185347x_{33} = 76.3849592185347
x33=7.12125046689807x_{33} = 7.12125046689807
x33=−77.5270825679419x_{33} = -77.5270825679419
x33=−64.9582857893902x_{33} = -64.9582857893902
x33=−33.5285657554621x_{33} = -33.5285657554621
x33=26.0929104121121x_{33} = 26.0929104121121
x33=−39.8162093266346x_{33} = -39.8162093266346
x33=38.672573565113x_{33} = 38.672573565113
x33=−46.1026627703624x_{33} = -46.1026627703624
Decrece en los intervalos
[98.3791034786112,∞)\left[98.3791034786112, \infty\right)
Crece en los intervalos
(−∞,−130.939312353727]\left(-\infty, -130.939312353727\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−1)+2sin⁡(x−1)x−1+2cos⁡(x−1)(x−1)2x−1=0\frac{- \cos{\left(x - 1 \right)} + \frac{2 \sin{\left(x - 1 \right)}}{x - 1} + \frac{2 \cos{\left(x - 1 \right)}}{\left(x - 1\right)^{2}}}{x - 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=37.0728437679879x_{1} = 37.0728437679879
x2=8.5873993379941x_{2} = 8.5873993379941
x3=59.085025007445x_{3} = 59.085025007445
x4=−68845.4321780434x_{4} = -68845.4321780434
x5=216.18980251639x_{5} = 216.18980251639
x6=96.7976970894915x_{6} = 96.7976970894915
x7=−25.6283591640252x_{7} = -25.6283591640252
x8=84.2281726832512x_{8} = 84.2281726832512
x9=−50.7976574095537x_{9} = -50.7976574095537
x10=−9.80950729816022x_{10} = -9.80950729816022
x11=−44.5091321154553x_{11} = -44.5091321154553
x12=99.9399529307048x_{12} = 99.9399529307048
x13=−31.9259431758392x_{13} = -31.9259431758392
x14=87.370639887736x_{14} = 87.370639887736
x15=68.5146145048817x_{15} = 68.5146145048817
x16=27.6283591640252x_{16} = 27.6283591640252
x17=81.0856368040887x_{17} = 81.0856368040887
x18=253.890299964068x_{18} = 253.890299964068
x19=−85.370639887736x_{19} = -85.370639887736
x20=−60.2283863503723x_{20} = -60.2283863503723
x21=43.3642737086586x_{21} = 43.3642737086586
x22=−75.9430238267933x_{22} = -75.9430238267933
x23=178.488716367408x_{23} = 178.488716367408
x24=93.655396245836x_{24} = 93.655396245836
x25=−22.4766510546492x_{25} = -22.4766510546492
x26=18.1619600917303x_{26} = 18.1619600917303
x27=65.3715747870554x_{27} = 65.3715747870554
x28=11.8095072981602x_{28} = 11.8095072981602
x29=−66.5146145048817x_{29} = -66.5146145048817
x30=−57.085025007445x_{30} = -57.085025007445
x31=−28.7779159141436x_{31} = -28.7779159141436
x32=49.6535676048409x_{32} = 49.6535676048409
x33=21.3217772482235x_{33} = 21.3217772482235
x34=−94.7976970894915x_{34} = -94.7976970894915
x35=−79.0856368040887x_{35} = -79.0856368040887
x36=40.218890250481x_{36} = 40.218890250481
x37=77.9430238267933x_{37} = 77.9430238267933
x38=−157.637821338313x_{38} = -157.637821338313
x39=−63.3715747870554x_{39} = -63.3715747870554
x40=24.4766510546492x_{40} = 24.4766510546492
x41=−16.1619600917303x_{41} = -16.1619600917303
x42=−97.9399529307048x_{42} = -97.9399529307048
x43=46.5091321154553x_{43} = 46.5091321154553
x44=52.7976574095537x_{44} = 52.7976574095537
x45=−82.2281726832512x_{45} = -82.2281726832512
x46=14.9937625671267x_{46} = 14.9937625671267
x47=71.6575253785884x_{47} = 71.6575253785884
x48=−91.655396245836x_{48} = -91.655396245836
x49=33.9259431758392x_{49} = 33.9259431758392
x50=−12.9937625671267x_{50} = -12.9937625671267
x51=90.5130456566371x_{51} = 90.5130456566371
x52=−101.082167928013x_{52} = -101.082167928013
x53=74.8003238908837x_{53} = 74.8003238908837
x54=−41.3642737086586x_{54} = -41.3642737086586
x55=−6.5873993379941x_{55} = -6.5873993379941
x56=62.2283863503723x_{56} = 62.2283863503723
x57=−3.2222763997912x_{57} = -3.2222763997912
x58=−69.6575253785884x_{58} = -69.6575253785884
x59=30.7779159141436x_{59} = 30.7779159141436
x60=−47.6535676048409x_{60} = -47.6535676048409
x61=−88.5130456566371x_{61} = -88.5130456566371
x62=−35.0728437679879x_{62} = -35.0728437679879
x63=5.2222763997912x_{63} = 5.2222763997912
x64=−38.218890250481x_{64} = -38.218890250481
x65=−19.3217772482235x_{65} = -19.3217772482235
x66=−72.8003238908837x_{66} = -72.8003238908837
x67=−53.9414610202918x_{67} = -53.9414610202918
x68=55.9414610202918x_{68} = 55.9414610202918
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=1x_{1} = 1

lim⁡x→1−(−cos⁡(x−1)+2sin⁡(x−1)x−1+2cos⁡(x−1)(x−1)2x−1)=−∞\lim_{x \to 1^-}\left(\frac{- \cos{\left(x - 1 \right)} + \frac{2 \sin{\left(x - 1 \right)}}{x - 1} + \frac{2 \cos{\left(x - 1 \right)}}{\left(x - 1\right)^{2}}}{x - 1}\right) = -\infty
lim⁡x→1+(−cos⁡(x−1)+2sin⁡(x−1)x−1+2cos⁡(x−1)(x−1)2x−1)=∞\lim_{x \to 1^+}\left(\frac{- \cos{\left(x - 1 \right)} + \frac{2 \sin{\left(x - 1 \right)}}{x - 1} + \frac{2 \cos{\left(x - 1 \right)}}{\left(x - 1\right)^{2}}}{x - 1}\right) = \infty
- los límites no son iguales, signo
x1=1x_{1} = 1
- 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
[253.890299964068,∞)\left[253.890299964068, \infty\right)
Convexa en los intervalos
(−∞,−68845.4321780434]\left(-\infty, -68845.4321780434\right]
Asíntotas verticales
Hay:
x1=1x_{1} = 1
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(cos⁡(x−1)x−1)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x - 1 \right)}}{x - 1}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(cos⁡(x−1)x−1)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x - 1 \right)}}{x - 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 cos(x - 1)/(x - 1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x−1)x(x−1))=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x - 1 \right)}}{x \left(x - 1\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x−1)x(x−1))=0\lim_{x \to \infty}\left(\frac{\cos{\left(x - 1 \right)}}{x \left(x - 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:
cos⁡(x−1)x−1=cos⁡(x+1)−x−1\frac{\cos{\left(x - 1 \right)}}{x - 1} = \frac{\cos{\left(x + 1 \right)}}{- x - 1}
- No
cos⁡(x−1)x−1=−cos⁡(x+1)−x−1\frac{\cos{\left(x - 1 \right)}}{x - 1} = - \frac{\cos{\left(x + 1 \right)}}{- x - 1}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор