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Trazado el gráfico de la función/ cosx/(x^2)

Gráfico de la función y = cosx/(x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x)
f(x) = ------
          2  
         x
f(x)=cos⁡(x)x2f{\left(x \right)} = \frac{\cos{\left(x \right)}}{x^{2}} 
f = cos(x)/x^2
Gráfico de la función
02468-8-6-4-2-1010-500500
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)x2=0\frac{\cos{\left(x \right)}}{x^{2}} = 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=14.1371669411541x_{1} = 14.1371669411541
x2=73.8274273593601x_{2} = 73.8274273593601
x3=−92.6769832808989x_{3} = -92.6769832808989
x4=−48.6946861306418x_{4} = -48.6946861306418
x5=186.924762888593x_{5} = 186.924762888593
x6=89.5353906273091x_{6} = 89.5353906273091
x7=−23.5619449019235x_{7} = -23.5619449019235
x8=−86.3937979737193x_{8} = -86.3937979737193
x9=39.2699081698724x_{9} = 39.2699081698724
x10=−17.2787595947439x_{10} = -17.2787595947439
x11=20.4203522483337x_{11} = 20.4203522483337
x12=−26.7035375555132x_{12} = -26.7035375555132
x13=61.261056745001x_{13} = 61.261056745001
x14=42.4115008234622x_{14} = 42.4115008234622
x15=−64.4026493985908x_{15} = -64.4026493985908
x16=−83.2522053201295x_{16} = -83.2522053201295
x17=−4.71238898038469x_{17} = -4.71238898038469
x18=−42.4115008234622x_{18} = -42.4115008234622
x19=−29.845130209103x_{19} = -29.845130209103
x20=17.2787595947439x_{20} = 17.2787595947439
x21=51.8362787842316x_{21} = 51.8362787842316
x22=1.5707963267949x_{22} = 1.5707963267949
x23=−67.5442420521806x_{23} = -67.5442420521806
x24=−36.1283155162826x_{24} = -36.1283155162826
x25=45.553093477052x_{25} = 45.553093477052
x26=−80.1106126665397x_{26} = -80.1106126665397
x27=86.3937979737193x_{27} = 86.3937979737193
x28=−73.8274273593601x_{28} = -73.8274273593601
x29=32.9867228626928x_{29} = 32.9867228626928
x30=64.4026493985908x_{30} = 64.4026493985908
x31=−1.5707963267949x_{31} = -1.5707963267949
x32=95.8185759344887x_{32} = 95.8185759344887
x33=−20.4203522483337x_{33} = -20.4203522483337
x34=−10.9955742875643x_{34} = -10.9955742875643
x35=−98.9601685880785x_{35} = -98.9601685880785
x36=92.6769832808989x_{36} = 92.6769832808989
x37=36.1283155162826x_{37} = 36.1283155162826
x38=−32.9867228626928x_{38} = -32.9867228626928
x39=−39.2699081698724x_{39} = -39.2699081698724
x40=−58.1194640914112x_{40} = -58.1194640914112
x41=−61.261056745001x_{41} = -61.261056745001
x42=4.71238898038469x_{42} = 4.71238898038469
x43=−76.9690200129499x_{43} = -76.9690200129499
x44=−95.8185759344887x_{44} = -95.8185759344887
x45=48.6946861306418x_{45} = 48.6946861306418
x46=−51.8362787842316x_{46} = -51.8362787842316
x47=23.5619449019235x_{47} = 23.5619449019235
x48=67.5442420521806x_{48} = 67.5442420521806
x49=−14.1371669411541x_{49} = -14.1371669411541
x50=76.9690200129499x_{50} = 76.9690200129499
x51=98.9601685880785x_{51} = 98.9601685880785
x52=80.1106126665397x_{52} = 80.1106126665397
x53=−7.85398163397448x_{53} = -7.85398163397448
x54=7.85398163397448x_{54} = 7.85398163397448
x55=83.2522053201295x_{55} = 83.2522053201295
x56=58.1194640914112x_{56} = 58.1194640914112
x57=−45.553093477052x_{57} = -45.553093477052
x58=70.6858347057703x_{58} = 70.6858347057703
x59=54.9778714378214x_{59} = 54.9778714378214
x60=26.7035375555132x_{60} = 26.7035375555132
x61=−89.5353906273091x_{61} = -89.5353906273091
x62=10.9955742875643x_{62} = 10.9955742875643
x63=−70.6858347057703x_{63} = -70.6858347057703
x64=−54.9778714378214x_{64} = -54.9778714378214
x65=29.845130209103x_{65} = 29.845130209103
x66=108.384946548848x_{66} = 108.384946548848
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^2.
cos⁡(0)02\frac{\cos{\left(0 \right)}}{0^{2}}
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)x2−2cos⁡(x)x3=0- \frac{\sin{\left(x \right)}}{x^{2}} - \frac{2 \cos{\left(x \right)}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=9.21096438740149x_{1} = 9.21096438740149
x2=81.6569211705466x_{2} = 81.6569211705466
x3=−78.5143487963623x_{3} = -78.5143487963623
x4=−37.6460352959305x_{4} = -37.6460352959305
x5=−15.5802941824244x_{5} = -15.5802941824244
x6=−84.7994209518635x_{6} = -84.7994209518635
x7=−91.0842327848165x_{7} = -91.0842327848165
x8=97.3688346960149x_{8} = 97.3688346960149
x9=25.053079662454x_{9} = 25.053079662454
x10=−2.45871417599962x_{10} = -2.45871417599962
x11=−72.2289483771681x_{11} = -72.2289483771681
x12=34.4996123350132x_{12} = 34.4996123350132
x13=−59.6567478435559x_{13} = -59.6567478435559
x14=31.3522215217643x_{14} = 31.3522215217643
x15=−9.21096438740149x_{15} = -9.21096438740149
x16=91.0842327848165x_{16} = 91.0842327848165
x17=75.3716947511882x_{17} = 75.3716947511882
x18=−69.0860970774096x_{18} = -69.0860970774096
x19=−5.95939190757933x_{19} = -5.95939190757933
x20=−12.4065403639626x_{20} = -12.4065403639626
x21=94.2265573558031x_{21} = 94.2265573558031
x22=−103.653264863525x_{22} = -103.653264863525
x23=37.6460352959305x_{23} = 37.6460352959305
x24=−34.4996123350132x_{24} = -34.4996123350132
x25=69.0860970774096x_{25} = 69.0860970774096
x26=−56.5132926241755x_{26} = -56.5132926241755
x27=12.4065403639626x_{27} = 12.4065403639626
x28=−75.3716947511882x_{28} = -75.3716947511882
x29=78.5143487963623x_{29} = 78.5143487963623
x30=56.5132926241755x_{30} = 56.5132926241755
x31=−109.93755273626x_{31} = -109.93755273626
x32=−21.9000773156394x_{32} = -21.9000773156394
x33=21.9000773156394x_{33} = 21.9000773156394
x34=87.9418559209576x_{34} = 87.9418559209576
x35=−87.9418559209576x_{35} = -87.9418559209576
x36=−43.9368086315937x_{36} = -43.9368086315937
x37=50.2256832197934x_{37} = 50.2256832197934
x38=−81.6569211705466x_{38} = -81.6569211705466
x39=40.7917141624847x_{39} = 40.7917141624847
x40=−31.3522215217643x_{40} = -31.3522215217643
x41=53.3696181339615x_{41} = 53.3696181339615
x42=−62.8000167068325x_{42} = -62.8000167068325
x43=−100.511069234565x_{43} = -100.511069234565
x44=−97.3688346960149x_{44} = -97.3688346960149
x45=−53.3696181339615x_{45} = -53.3696181339615
x46=−65.9431258539286x_{46} = -65.9431258539286
x47=18.7432530945386x_{47} = 18.7432530945386
x48=−47.0814357397523x_{48} = -47.0814357397523
x49=15.5802941824244x_{49} = 15.5802941824244
x50=100.511069234565x_{50} = 100.511069234565
x51=43.9368086315937x_{51} = 43.9368086315937
x52=65.9431258539286x_{52} = 65.9431258539286
x53=−18.7432530945386x_{53} = -18.7432530945386
x54=135.073678452493x_{54} = 135.073678452493
x55=−28.2035393053095x_{55} = -28.2035393053095
x56=62.8000167068325x_{56} = 62.8000167068325
x57=2.45871417599962x_{57} = 2.45871417599962
x58=−25.053079662454x_{58} = -25.053079662454
x59=−94.2265573558031x_{59} = -94.2265573558031
x60=84.7994209518635x_{60} = 84.7994209518635
x61=72.2289483771681x_{61} = 72.2289483771681
x62=−40.7917141624847x_{62} = -40.7917141624847
x63=47.0814357397523x_{63} = 47.0814357397523
x64=59.6567478435559x_{64} = 59.6567478435559
x65=28.2035393053095x_{65} = 28.2035393053095
x66=−50.2256832197934x_{66} = -50.2256832197934
x67=5.95939190757933x_{67} = 5.95939190757933
Signos de extremos en los puntos:
(9.210964387401486, -0.0115182384102548)

(81.65692117054658, 0.000149928353545869)

(-78.51434879636227, -0.000162166475547147)

(-37.64603529593052, 0.000704611119614408)

(-15.580294182424433, -0.00408601227579287)

(-84.79942095186354, -0.000139025181535869)

(-91.08423278481655, -0.000120506069272649)

(97.36883469601494, -0.000105455311245964)

(25.053079662453992, 0.00158817477024791)

(-2.4587141759996247, -0.128324928485094)

(-72.22894837716808, -0.00019160683134921)

(34.4996123350132, -0.000838770260526343)

(-59.656747843555884, -0.000280825751144458)

(31.352221521764292, 0.0010152698990766)

(-9.210964387401486, -0.0115182384102548)

(91.08423278481655, -0.000120506069272649)

(75.37169475118824, 0.000175966743144092)

(-69.08609707740959, 0.000209428978902002)

(-5.9593919075793265, 0.0266944281300046)

(-12.406540363962565, 0.00641398077993427)

(94.22655735580307, 0.000112604447700661)

(-103.65326486352524, -9.30578895300905e-5)

(37.64603529593052, 0.000704611119614408)

(-34.4996123350132, -0.000838770260526343)

(69.08609707740959, 0.000209428978902002)

(-56.513292624175506, 0.000312915432650295)

(12.406540363962565, 0.00641398077993427)

(-75.37169475118824, 0.000175966743144092)

(78.51434879636227, -0.000162166475547147)

(56.513292624175506, 0.000312915432650295)

(-109.93755273625987, -8.27248552367837e-5)

(-21.90007731563936, -0.00207637214990232)

(21.90007731563936, -0.00207637214990232)

(87.94185592095755, 0.000129269617694298)

(-87.94185592095755, 0.000129269617694298)

(-43.936808631593706, 0.000517479923876906)

(50.2256832197934, 0.000396099456126142)

(-81.65692117054658, 0.000149928353545869)

(40.79171416248471, -0.00060025351930421)

(-31.352221521764292, 0.0010152698990766)

(53.36961813396146, -0.000350838362181669)

(-62.80001670683253, 0.000253431359776371)

(-100.51106923456473, 9.89660537297585e-5)

(-97.36883469601494, -0.000105455311245964)

(-53.36961813396146, -0.000350838362181669)

(-65.94312585392862, -0.000229858880631)

(18.74325309453857, 0.00283042465312132)

(-47.081435739752315, -0.000450722368032648)

(15.580294182424433, -0.00408601227579287)

(100.51106923456473, 9.89660537297585e-5)

(43.936808631593706, 0.000517479923876906)

(65.94312585392862, -0.000229858880631)

(-18.74325309453857, 0.00283042465312132)

(135.07367845249348, -5.48038341935653e-5)

(-28.20353930530947, -0.00125401736797822)

(62.80001670683253, 0.000253431359776371)

(2.4587141759996247, -0.128324928485094)

(-25.053079662453992, 0.00158817477024791)

(-94.22655735580307, 0.000112604447700661)

(84.79942095186354, -0.000139025181535869)

(72.22894837716808, -0.00019160683134921)

(-40.79171416248471, -0.00060025351930421)

(47.081435739752315, -0.000450722368032648)

(59.656747843555884, -0.000280825751144458)

(28.20353930530947, -0.00125401736797822)

(-50.2256832197934, 0.000396099456126142)

(5.9593919075793265, 0.0266944281300046)


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=9.21096438740149x_{1} = 9.21096438740149
x2=−78.5143487963623x_{2} = -78.5143487963623
x3=−15.5802941824244x_{3} = -15.5802941824244
x4=−84.7994209518635x_{4} = -84.7994209518635
x5=−91.0842327848165x_{5} = -91.0842327848165
x6=97.3688346960149x_{6} = 97.3688346960149
x7=−2.45871417599962x_{7} = -2.45871417599962
x8=−72.2289483771681x_{8} = -72.2289483771681
x9=34.4996123350132x_{9} = 34.4996123350132
x10=−59.6567478435559x_{10} = -59.6567478435559
x11=−9.21096438740149x_{11} = -9.21096438740149
x12=91.0842327848165x_{12} = 91.0842327848165
x13=−103.653264863525x_{13} = -103.653264863525
x14=−34.4996123350132x_{14} = -34.4996123350132
x15=78.5143487963623x_{15} = 78.5143487963623
x16=−109.93755273626x_{16} = -109.93755273626
x17=−21.9000773156394x_{17} = -21.9000773156394
x18=21.9000773156394x_{18} = 21.9000773156394
x19=40.7917141624847x_{19} = 40.7917141624847
x20=53.3696181339615x_{20} = 53.3696181339615
x21=−97.3688346960149x_{21} = -97.3688346960149
x22=−53.3696181339615x_{22} = -53.3696181339615
x23=−65.9431258539286x_{23} = -65.9431258539286
x24=−47.0814357397523x_{24} = -47.0814357397523
x25=15.5802941824244x_{25} = 15.5802941824244
x26=65.9431258539286x_{26} = 65.9431258539286
x27=135.073678452493x_{27} = 135.073678452493
x28=−28.2035393053095x_{28} = -28.2035393053095
x29=2.45871417599962x_{29} = 2.45871417599962
x30=84.7994209518635x_{30} = 84.7994209518635
x31=72.2289483771681x_{31} = 72.2289483771681
x32=−40.7917141624847x_{32} = -40.7917141624847
x33=47.0814357397523x_{33} = 47.0814357397523
x34=59.6567478435559x_{34} = 59.6567478435559
x35=28.2035393053095x_{35} = 28.2035393053095
Puntos máximos de la función:
x35=81.6569211705466x_{35} = 81.6569211705466
x35=−37.6460352959305x_{35} = -37.6460352959305
x35=25.053079662454x_{35} = 25.053079662454
x35=31.3522215217643x_{35} = 31.3522215217643
x35=75.3716947511882x_{35} = 75.3716947511882
x35=−69.0860970774096x_{35} = -69.0860970774096
x35=−5.95939190757933x_{35} = -5.95939190757933
x35=−12.4065403639626x_{35} = -12.4065403639626
x35=94.2265573558031x_{35} = 94.2265573558031
x35=37.6460352959305x_{35} = 37.6460352959305
x35=69.0860970774096x_{35} = 69.0860970774096
x35=−56.5132926241755x_{35} = -56.5132926241755
x35=12.4065403639626x_{35} = 12.4065403639626
x35=−75.3716947511882x_{35} = -75.3716947511882
x35=56.5132926241755x_{35} = 56.5132926241755
x35=87.9418559209576x_{35} = 87.9418559209576
x35=−87.9418559209576x_{35} = -87.9418559209576
x35=−43.9368086315937x_{35} = -43.9368086315937
x35=50.2256832197934x_{35} = 50.2256832197934
x35=−81.6569211705466x_{35} = -81.6569211705466
x35=−31.3522215217643x_{35} = -31.3522215217643
x35=−62.8000167068325x_{35} = -62.8000167068325
x35=−100.511069234565x_{35} = -100.511069234565
x35=18.7432530945386x_{35} = 18.7432530945386
x35=100.511069234565x_{35} = 100.511069234565
x35=43.9368086315937x_{35} = 43.9368086315937
x35=−18.7432530945386x_{35} = -18.7432530945386
x35=62.8000167068325x_{35} = 62.8000167068325
x35=−25.053079662454x_{35} = -25.053079662454
x35=−94.2265573558031x_{35} = -94.2265573558031
x35=−50.2256832197934x_{35} = -50.2256832197934
x35=5.95939190757933x_{35} = 5.95939190757933
Decrece en los intervalos
[135.073678452493,∞)\left[135.073678452493, \infty\right)
Crece en los intervalos
(−∞,−109.93755273626]\left(-\infty, -109.93755273626\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)+4sin⁡(x)x+6cos⁡(x)x2x2=0\frac{- \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{x} + \frac{6 \cos{\left(x \right)}}{x^{2}}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=54.9050022900618x_{1} = 54.9050022900618
x2=−61.1956810599968x_{2} = -61.1956810599968
x3=7.3008858832282x_{3} = 7.3008858832282
x4=39.167739309003x_{4} = 39.167739309003
x5=−20.2222381068518x_{5} = -20.2222381068518
x6=−45.4650856843323x_{6} = -45.4650856843323
x7=−76.9170100644479x_{7} = -76.9170100644479
x8=83.2041261605509x_{8} = 83.2041261605509
x9=−86.3474693776984x_{9} = -86.3474693776984
x10=−124.060666131211x_{10} = -124.060666131211
x11=13.8473675657494x_{11} = 13.8473675657494
x12=−58.0505450446381x_{12} = -58.0505450446381
x13=20.2222381068518x_{13} = 20.2222381068518
x14=−92.6337991480858x_{14} = -92.6337991480858
x15=61.1956810599968x_{15} = 61.1956810599968
x16=42.3169411013932x_{16} = 42.3169411013932
x17=−32.8649384028032x_{17} = -32.8649384028032
x18=−73.7732005024933x_{18} = -73.7732005024933
x19=17.0435524410056x_{19} = 17.0435524410056
x20=−83.2041261605509x_{20} = -83.2041261605509
x21=26.5527542671168x_{21} = 26.5527542671168
x22=70.6291933516676x_{22} = 70.6291933516676
x23=10.6168428630359x_{23} = 10.6168428630359
x24=−89.4906895254562x_{24} = -89.4906895254562
x25=67.4849609355352x_{25} = 67.4849609355352
x26=3.58835703497019x_{26} = 3.58835703497019
x27=36.0172009751693x_{27} = 36.0172009751693
x28=23.3907336526907x_{28} = 23.3907336526907
x29=−80.0606453553676x_{29} = -80.0606453553676
x30=58.0505450446381x_{30} = 58.0505450446381
x31=89.4906895254562x_{31} = 89.4906895254562
x32=−95.7768091402129x_{32} = -95.7768091402129
x33=−13.8473675657494x_{33} = -13.8473675657494
x34=−29.7103970410056x_{34} = -29.7103970410056
x35=−10.6168428630359x_{35} = -10.6168428630359
x36=−48.6123794800497x_{36} = -48.6123794800497
x37=−39.167739309003x_{37} = -39.167739309003
x38=−26.5527542671168x_{38} = -26.5527542671168
x39=397.401405246951x_{39} = 397.401405246951
x40=48.6123794800497x_{40} = 48.6123794800497
x41=−64.3404701495554x_{41} = -64.3404701495554
x42=318.859109531625x_{42} = 318.859109531625
x43=105.205330720799x_{43} = 105.205330720799
x44=−54.9050022900618x_{44} = -54.9050022900618
x45=29.7103970410056x_{45} = 29.7103970410056
x46=45.4650856843323x_{46} = 45.4650856843323
x47=76.9170100644479x_{47} = 76.9170100644479
x48=32.8649384028032x_{48} = 32.8649384028032
x49=51.7589783694261x_{49} = 51.7589783694261
x50=−98.9197290094896x_{50} = -98.9197290094896
x51=92.6337991480858x_{51} = 92.6337991480858
x52=−51.7589783694261x_{52} = -51.7589783694261
x53=−36.0172009751693x_{53} = -36.0172009751693
x54=−7.3008858832282x_{54} = -7.3008858832282
x55=80.0606453553676x_{55} = 80.0606453553676
x56=64.3404701495554x_{56} = 64.3404701495554
x57=95.7768091402129x_{57} = 95.7768091402129
x58=−17.0435524410056x_{58} = -17.0435524410056
x59=−42.3169411013932x_{59} = -42.3169411013932
x60=73.7732005024933x_{60} = 73.7732005024933
x61=−3.58835703497019x_{61} = -3.58835703497019
x62=86.3474693776984x_{62} = 86.3474693776984
x63=−23.3907336526907x_{63} = -23.3907336526907
x64=−67.4849609355352x_{64} = -67.4849609355352
x65=−70.6291933516676x_{65} = -70.6291933516676
x66=98.9197290094896x_{66} = 98.9197290094896
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)+4sin⁡(x)x+6cos⁡(x)x2x2)=∞\lim_{x \to 0^-}\left(\frac{- \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{x} + \frac{6 \cos{\left(x \right)}}{x^{2}}}{x^{2}}\right) = \infty
lim⁡x→0+(−cos⁡(x)+4sin⁡(x)x+6cos⁡(x)x2x2)=∞\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{x} + \frac{6 \cos{\left(x \right)}}{x^{2}}}{x^{2}}\right) = \infty
- 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
[397.401405246951,∞)\left[397.401405246951, \infty\right)
Convexa en los intervalos
(−∞,−124.060666131211]\left(-\infty, -124.060666131211\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)x2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
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,
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^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x)xx2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x)xx2)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x 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)x2=cos⁡(x)x2\frac{\cos{\left(x \right)}}{x^{2}} = \frac{\cos{\left(x \right)}}{x^{2}}
- Sí
cos⁡(x)x2=−cos⁡(x)x2\frac{\cos{\left(x \right)}}{x^{2}} = - \frac{\cos{\left(x \right)}}{x^{2}}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор