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

Gráfico de la función y = cos(x)/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

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

Solución analítica
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Solución numérica
x1=32.9867228626928x_{1} = 32.9867228626928
x2=73.8274273593601x_{2} = 73.8274273593601
x3=4.71238898038469x_{3} = 4.71238898038469
x4=39.2699081698724x_{4} = 39.2699081698724
x5=95.8185759344887x_{5} = 95.8185759344887
x6=45.553093477052x_{6} = 45.553093477052
x7=70.6858347057703x_{7} = 70.6858347057703
x8=−10.9955742875643x_{8} = -10.9955742875643
x9=−58.1194640914112x_{9} = -58.1194640914112
x10=−23.5619449019235x_{10} = -23.5619449019235
x11=199.491133502952x_{11} = 199.491133502952
x12=−26.7035375555132x_{12} = -26.7035375555132
x13=347.145988221672x_{13} = 347.145988221672
x14=−89.5353906273091x_{14} = -89.5353906273091
x15=−17.2787595947439x_{15} = -17.2787595947439
x16=26.7035375555132x_{16} = 26.7035375555132
x17=−42.4115008234622x_{17} = -42.4115008234622
x18=−61.261056745001x_{18} = -61.261056745001
x19=92.6769832808989x_{19} = 92.6769832808989
x20=−76.9690200129499x_{20} = -76.9690200129499
x21=−92.6769832808989x_{21} = -92.6769832808989
x22=−98.9601685880785x_{22} = -98.9601685880785
x23=61.261056745001x_{23} = 61.261056745001
x24=−54.9778714378214x_{24} = -54.9778714378214
x25=42.4115008234622x_{25} = 42.4115008234622
x26=−422.544211907827x_{26} = -422.544211907827
x27=−64.4026493985908x_{27} = -64.4026493985908
x28=67.5442420521806x_{28} = 67.5442420521806
x29=−7.85398163397448x_{29} = -7.85398163397448
x30=80.1106126665397x_{30} = 80.1106126665397
x31=−14.1371669411541x_{31} = -14.1371669411541
x32=14.1371669411541x_{32} = 14.1371669411541
x33=1173.38485611579x_{33} = 1173.38485611579
x34=−1.5707963267949x_{34} = -1.5707963267949
x35=1.5707963267949x_{35} = 1.5707963267949
x36=29.845130209103x_{36} = 29.845130209103
x37=10.9955742875643x_{37} = 10.9955742875643
x38=17.2787595947439x_{38} = 17.2787595947439
x39=−51.8362787842316x_{39} = -51.8362787842316
x40=−29.845130209103x_{40} = -29.845130209103
x41=−48.6946861306418x_{41} = -48.6946861306418
x42=−73.8274273593601x_{42} = -73.8274273593601
x43=23.5619449019235x_{43} = 23.5619449019235
x44=20.4203522483337x_{44} = 20.4203522483337
x45=−86.3937979737193x_{45} = -86.3937979737193
x46=54.9778714378214x_{46} = 54.9778714378214
x47=58.1194640914112x_{47} = 58.1194640914112
x48=51.8362787842316x_{48} = 51.8362787842316
x49=−67.5442420521806x_{49} = -67.5442420521806
x50=−4.71238898038469x_{50} = -4.71238898038469
x51=−45.553093477052x_{51} = -45.553093477052
x52=−70.6858347057703x_{52} = -70.6858347057703
x53=48.6946861306418x_{53} = 48.6946861306418
x54=−83.2522053201295x_{54} = -83.2522053201295
x55=−95.8185759344887x_{55} = -95.8185759344887
x56=89.5353906273091x_{56} = 89.5353906273091
x57=−39.2699081698724x_{57} = -39.2699081698724
x58=76.9690200129499x_{58} = 76.9690200129499
x59=−32.9867228626928x_{59} = -32.9867228626928
x60=−20.4203522483337x_{60} = -20.4203522483337
x61=−36.1283155162826x_{61} = -36.1283155162826
x62=7.85398163397448x_{62} = 7.85398163397448
x63=−80.1106126665397x_{63} = -80.1106126665397
x64=86.3937979737193x_{64} = 86.3937979737193
x65=98.9601685880785x_{65} = 98.9601685880785
x66=36.1283155162826x_{66} = 36.1283155162826
x67=64.4026493985908x_{67} = 64.4026493985908
x68=83.2522053201295x_{68} = 83.2522053201295
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)/x.
cos⁡(0)0\frac{\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)x2=0- \frac{\sin{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=97.3791034786112x_{1} = 97.3791034786112
x2=50.2455828375744x_{2} = 50.2455828375744
x3=59.6735041304405x_{3} = 59.6735041304405
x4=28.2389365752603x_{4} = 28.2389365752603
x5=−91.0952098694071x_{5} = -91.0952098694071
x6=−43.9595528888955x_{6} = -43.9595528888955
x7=−47.1026627703624x_{7} = -47.1026627703624
x8=−75.3849592185347x_{8} = -75.3849592185347
x9=−135.08108127842x_{9} = -135.08108127842
x10=−78.5270825679419x_{10} = -78.5270825679419
x11=−56.5309801938186x_{11} = -56.5309801938186
x12=−94.2371684817036x_{12} = -94.2371684817036
x13=−25.0929104121121x_{13} = -25.0929104121121
x14=37.672573565113x_{14} = 37.672573565113
x15=40.8162093266346x_{15} = 40.8162093266346
x16=53.3883466217256x_{16} = 53.3883466217256
x17=65.9582857893902x_{17} = 65.9582857893902
x18=−40.8162093266346x_{18} = -40.8162093266346
x19=−169.640108529775x_{19} = -169.640108529775
x20=−34.5285657554621x_{20} = -34.5285657554621
x21=69.100567727981x_{21} = 69.100567727981
x22=−109.946647805931x_{22} = -109.946647805931
x23=34.5285657554621x_{23} = 34.5285657554621
x24=81.6691650818489x_{24} = 81.6691650818489
x25=84.811211299318x_{25} = 84.811211299318
x26=−81.6691650818489x_{26} = -81.6691650818489
x27=−37.672573565113x_{27} = -37.672573565113
x28=18.7964043662102x_{28} = 18.7964043662102
x29=−62.8159348889734x_{29} = -62.8159348889734
x30=197.91528455229x_{30} = 197.91528455229
x31=25.0929104121121x_{31} = 25.0929104121121
x32=2.79838604578389x_{32} = 2.79838604578389
x33=87.9532251106725x_{33} = 87.9532251106725
x34=−9.31786646179107x_{34} = -9.31786646179107
x35=−12.4864543952238x_{35} = -12.4864543952238
x36=−84.811211299318x_{36} = -84.811211299318
x37=−50.2455828375744x_{37} = -50.2455828375744
x38=−21.945612879981x_{38} = -21.945612879981
x39=−100.521017074687x_{39} = -100.521017074687
x40=−97.3791034786112x_{40} = -97.3791034786112
x41=−6.12125046689807x_{41} = -6.12125046689807
x42=−18.7964043662102x_{42} = -18.7964043662102
x43=43.9595528888955x_{43} = 43.9595528888955
x44=100.521017074687x_{44} = 100.521017074687
x45=31.3840740178899x_{45} = 31.3840740178899
x46=−65.9582857893902x_{46} = -65.9582857893902
x47=72.2427897046973x_{47} = 72.2427897046973
x48=94.2371684817036x_{48} = 94.2371684817036
x49=78.5270825679419x_{49} = 78.5270825679419
x50=47.1026627703624x_{50} = 47.1026627703624
x51=−87.9532251106725x_{51} = -87.9532251106725
x52=−15.644128370333x_{52} = -15.644128370333
x53=75.3849592185347x_{53} = 75.3849592185347
x54=62.8159348889734x_{54} = 62.8159348889734
x55=−28.2389365752603x_{55} = -28.2389365752603
x56=−31.3840740178899x_{56} = -31.3840740178899
x57=15.644128370333x_{57} = 15.644128370333
x58=−72.2427897046973x_{58} = -72.2427897046973
x59=56.5309801938186x_{59} = 56.5309801938186
x60=9.31786646179107x_{60} = 9.31786646179107
x61=−53.3883466217256x_{61} = -53.3883466217256
x62=6.12125046689807x_{62} = 6.12125046689807
x63=−69.100567727981x_{63} = -69.100567727981
x64=−2.79838604578389x_{64} = -2.79838604578389
x65=−59.6735041304405x_{65} = -59.6735041304405
x66=91.0952098694071x_{66} = 91.0952098694071
x67=21.945612879981x_{67} = 21.945612879981
x68=12.4864543952238x_{68} = 12.4864543952238
Signos de extremos en los puntos:
(97.3791034786112, -0.0102686022030809)

(50.24558283757444, 0.0198983065303553)

(59.67350413044053, -0.0167555036571887)

(28.238936575260272, -0.0353899155541688)

(-91.09520986940714, 0.0109768642483425)

(-43.959552888895495, -0.0227423004725314)

(-47.10266277036235, 0.0212254394164143)

(-75.38495921853475, -0.0132640786518247)

(-135.0810812784199, 0.00740275832666827)

(-78.52708256794193, 0.0127334276777468)

(-56.53098019381864, -0.0176866485521696)

(-94.23716848170359, -0.01061092686295)

(-25.092910412112097, -0.0398202855500511)

(37.67257356511297, 0.0265351630103045)

(40.81620932663458, -0.0244927205346957)

(53.38834662172563, -0.0187273944640866)

(65.95828578939016, -0.0151593553168405)

(-40.81620932663458, 0.0244927205346957)

(-169.6401085297751, -0.00589472993500857)

(-34.52856575546206, 0.0289493889114503)

(69.10056772798097, 0.0144701459746764)

(-109.94664780593057, 0.00909494432157336)

(34.52856575546206, -0.0289493889114503)

(81.66916508184887, 0.0122436055670467)

(84.81121129931802, -0.0117900744410766)

(-81.66916508184887, -0.0122436055670467)

(-37.67257356511297, -0.0265351630103045)

(18.796404366210158, 0.0531265325613881)

(-62.81593488897342, -0.015917510583426)

(197.91528455229027, -0.00505260236866135)

(25.092910412112097, 0.0398202855500511)

(2.798386045783887, -0.336508416918395)

(87.95322511067255, 0.0113689449158811)

(-9.317866461791066, 0.106707947715237)

(-12.486454395223781, -0.0798311807800032)

(-84.81121129931802, 0.0117900744410766)

(-50.24558283757444, -0.0198983065303553)

(-21.945612879981045, 0.0455199604051285)

(-100.52101707468658, -0.00994767611536293)

(-97.3791034786112, 0.0102686022030809)

(-6.1212504668980685, -0.161228034325064)

(-18.796404366210158, -0.0531265325613881)

(43.959552888895495, 0.0227423004725314)

(100.52101707468658, 0.00994767611536293)

(31.38407401788986, 0.0318471321112693)

(-65.95828578939016, 0.0151593553168405)

(72.24278970469729, -0.0138408859131547)

(94.23716848170359, 0.01061092686295)

(78.52708256794193, -0.0127334276777468)

(47.10266277036235, -0.0212254394164143)

(-87.95322511067255, -0.0113689449158811)

(-15.644128370333028, 0.0637915530395936)

(75.38495921853475, 0.0132640786518247)

(62.81593488897342, 0.015917510583426)

(-28.238936575260272, 0.0353899155541688)

(-31.38407401788986, -0.0318471321112693)

(15.644128370333028, -0.0637915530395936)

(-72.24278970469729, 0.0138408859131547)

(56.53098019381864, 0.0176866485521696)

(9.317866461791066, -0.106707947715237)

(-53.38834662172563, 0.0187273944640866)

(6.1212504668980685, 0.161228034325064)

(-69.10056772798097, -0.0144701459746764)

(-2.798386045783887, 0.336508416918395)

(-59.67350413044053, 0.0167555036571887)

(91.09520986940714, -0.0109768642483425)

(21.945612879981045, -0.0455199604051285)

(12.486454395223781, 0.0798311807800032)


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=97.3791034786112x_{1} = 97.3791034786112
x2=59.6735041304405x_{2} = 59.6735041304405
x3=28.2389365752603x_{3} = 28.2389365752603
x4=−43.9595528888955x_{4} = -43.9595528888955
x5=−75.3849592185347x_{5} = -75.3849592185347
x6=−56.5309801938186x_{6} = -56.5309801938186
x7=−94.2371684817036x_{7} = -94.2371684817036
x8=−25.0929104121121x_{8} = -25.0929104121121
x9=40.8162093266346x_{9} = 40.8162093266346
x10=53.3883466217256x_{10} = 53.3883466217256
x11=65.9582857893902x_{11} = 65.9582857893902
x12=−169.640108529775x_{12} = -169.640108529775
x13=34.5285657554621x_{13} = 34.5285657554621
x14=84.811211299318x_{14} = 84.811211299318
x15=−81.6691650818489x_{15} = -81.6691650818489
x16=−37.672573565113x_{16} = -37.672573565113
x17=−62.8159348889734x_{17} = -62.8159348889734
x18=197.91528455229x_{18} = 197.91528455229
x19=2.79838604578389x_{19} = 2.79838604578389
x20=−12.4864543952238x_{20} = -12.4864543952238
x21=−50.2455828375744x_{21} = -50.2455828375744
x22=−100.521017074687x_{22} = -100.521017074687
x23=−6.12125046689807x_{23} = -6.12125046689807
x24=−18.7964043662102x_{24} = -18.7964043662102
x25=72.2427897046973x_{25} = 72.2427897046973
x26=78.5270825679419x_{26} = 78.5270825679419
x27=47.1026627703624x_{27} = 47.1026627703624
x28=−87.9532251106725x_{28} = -87.9532251106725
x29=−31.3840740178899x_{29} = -31.3840740178899
x30=15.644128370333x_{30} = 15.644128370333
x31=9.31786646179107x_{31} = 9.31786646179107
x32=−69.100567727981x_{32} = -69.100567727981
x33=91.0952098694071x_{33} = 91.0952098694071
x34=21.945612879981x_{34} = 21.945612879981
Puntos máximos de la función:
x34=50.2455828375744x_{34} = 50.2455828375744
x34=−91.0952098694071x_{34} = -91.0952098694071
x34=−47.1026627703624x_{34} = -47.1026627703624
x34=−135.08108127842x_{34} = -135.08108127842
x34=−78.5270825679419x_{34} = -78.5270825679419
x34=37.672573565113x_{34} = 37.672573565113
x34=−40.8162093266346x_{34} = -40.8162093266346
x34=−34.5285657554621x_{34} = -34.5285657554621
x34=69.100567727981x_{34} = 69.100567727981
x34=−109.946647805931x_{34} = -109.946647805931
x34=81.6691650818489x_{34} = 81.6691650818489
x34=18.7964043662102x_{34} = 18.7964043662102
x34=25.0929104121121x_{34} = 25.0929104121121
x34=87.9532251106725x_{34} = 87.9532251106725
x34=−9.31786646179107x_{34} = -9.31786646179107
x34=−84.811211299318x_{34} = -84.811211299318
x34=−21.945612879981x_{34} = -21.945612879981
x34=−97.3791034786112x_{34} = -97.3791034786112
x34=43.9595528888955x_{34} = 43.9595528888955
x34=100.521017074687x_{34} = 100.521017074687
x34=31.3840740178899x_{34} = 31.3840740178899
x34=−65.9582857893902x_{34} = -65.9582857893902
x34=94.2371684817036x_{34} = 94.2371684817036
x34=−15.644128370333x_{34} = -15.644128370333
x34=75.3849592185347x_{34} = 75.3849592185347
x34=62.8159348889734x_{34} = 62.8159348889734
x34=−28.2389365752603x_{34} = -28.2389365752603
x34=−72.2427897046973x_{34} = -72.2427897046973
x34=56.5309801938186x_{34} = 56.5309801938186
x34=−53.3883466217256x_{34} = -53.3883466217256
x34=6.12125046689807x_{34} = 6.12125046689807
x34=−2.79838604578389x_{34} = -2.79838604578389
x34=−59.6735041304405x_{34} = -59.6735041304405
x34=12.4864543952238x_{34} = 12.4864543952238
Decrece en los intervalos
[197.91528455229,∞)\left[197.91528455229, \infty\right)
Crece en los intervalos
(−∞,−169.640108529775]\left(-\infty, -169.640108529775\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+2cos⁡(x)x2x=0\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \right)}}{x^{2}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−51.7976574095537x_{1} = -51.7976574095537
x2=54.9414610202918x_{2} = 54.9414610202918
x3=−48.6535676048409x_{3} = -48.6535676048409
x4=4.2222763997912x_{4} = 4.2222763997912
x5=−32.9259431758392x_{5} = -32.9259431758392
x6=−23.4766510546492x_{6} = -23.4766510546492
x7=89.5130456566371x_{7} = 89.5130456566371
x8=32.9259431758392x_{8} = 32.9259431758392
x9=36.0728437679879x_{9} = 36.0728437679879
x10=−45.5091321154553x_{10} = -45.5091321154553
x11=17.1619600917303x_{11} = 17.1619600917303
x12=−10.8095072981602x_{12} = -10.8095072981602
x13=−26.6283591640252x_{13} = -26.6283591640252
x14=−17.1619600917303x_{14} = -17.1619600917303
x15=−42.3642737086586x_{15} = -42.3642737086586
x16=48.6535676048409x_{16} = 48.6535676048409
x17=−67.5146145048817x_{17} = -67.5146145048817
x18=86.370639887736x_{18} = 86.370639887736
x19=7.5873993379941x_{19} = 7.5873993379941
x20=70.6575253785884x_{20} = 70.6575253785884
x21=95.7976970894915x_{21} = 95.7976970894915
x22=−29.7779159141436x_{22} = -29.7779159141436
x23=51.7976574095537x_{23} = 51.7976574095537
x24=−61.2283863503723x_{24} = -61.2283863503723
x25=−95.7976970894915x_{25} = -95.7976970894915
x26=29.7779159141436x_{26} = 29.7779159141436
x27=83.2281726832512x_{27} = 83.2281726832512
x28=−36.0728437679879x_{28} = -36.0728437679879
x29=−13.9937625671267x_{29} = -13.9937625671267
x30=23.4766510546492x_{30} = 23.4766510546492
x31=−76.9430238267933x_{31} = -76.9430238267933
x32=−20.3217772482235x_{32} = -20.3217772482235
x33=80.0856368040887x_{33} = 80.0856368040887
x34=−80.0856368040887x_{34} = -80.0856368040887
x35=26.6283591640252x_{35} = 26.6283591640252
x36=−92.655396245836x_{36} = -92.655396245836
x37=−271.740404503579x_{37} = -271.740404503579
x38=58.085025007445x_{38} = 58.085025007445
x39=98.9399529307048x_{39} = 98.9399529307048
x40=−4.2222763997912x_{40} = -4.2222763997912
x41=−70.6575253785884x_{41} = -70.6575253785884
x42=230.898398112111x_{42} = 230.898398112111
x43=−54.9414610202918x_{43} = -54.9414610202918
x44=64.3715747870554x_{44} = 64.3715747870554
x45=45.5091321154553x_{45} = 45.5091321154553
x46=−64.3715747870554x_{46} = -64.3715747870554
x47=−73.8003238908837x_{47} = -73.8003238908837
x48=−83.2281726832512x_{48} = -83.2281726832512
x49=76.9430238267933x_{49} = 76.9430238267933
x50=73.8003238908837x_{50} = 73.8003238908837
x51=42.3642737086586x_{51} = 42.3642737086586
x52=61.2283863503723x_{52} = 61.2283863503723
x53=−39.218890250481x_{53} = -39.218890250481
x54=20.3217772482235x_{54} = 20.3217772482235
x55=−7.5873993379941x_{55} = -7.5873993379941
x56=10.8095072981602x_{56} = 10.8095072981602
x57=−86.370639887736x_{57} = -86.370639887736
x58=−58.085025007445x_{58} = -58.085025007445
x59=−98.9399529307048x_{59} = -98.9399529307048
x60=13.9937625671267x_{60} = 13.9937625671267
x61=−89.5130456566371x_{61} = -89.5130456566371
x62=67.5146145048817x_{62} = 67.5146145048817
x63=92.655396245836x_{63} = 92.655396245836
x64=39.218890250481x_{64} = 39.218890250481
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+2cos⁡(x)x2x)=−∞\lim_{x \to 0^-}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \right)}}{x^{2}}}{x}\right) = -\infty
lim⁡x→0+(−cos⁡(x)+2sin⁡(x)x+2cos⁡(x)x2x)=∞\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \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
[95.7976970894915,∞)\left[95.7976970894915, \infty\right)
Convexa en los intervalos
(−∞,−271.740404503579]\left(-\infty, -271.740404503579\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)x)=0\lim_{x \to -\infty}\left(\frac{\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→∞(cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\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 cos(x)/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x)x2)=0\lim_{x \to -\infty}\left(\frac{\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→∞(cos⁡(x)x2)=0\lim_{x \to \infty}\left(\frac{\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:
cos⁡(x)x=−cos⁡(x)x\frac{\cos{\left(x \right)}}{x} = - \frac{\cos{\left(x \right)}}{x}
- No
cos⁡(x)x=cos⁡(x)x\frac{\cos{\left(x \right)}}{x} = \frac{\cos{\left(x \right)}}{x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = cos(x)/x
    © Sr Examen – Калькулятор