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cos(x)/(x^2+1)
Trazado el gráfico de la función/ cos(x)/(x^2+1)

Gráfico de la función y = cos(x)/(x^2+1)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x)
f(x) = ------
        2    
       x  + 1
f(x)=cos⁡(x)x2+1f{\left(x \right)} = \frac{\cos{\left(x \right)}}{x^{2} + 1} 
f = cos(x)/(x^2 + 1)
Gráfico de la función
02468-8-6-4-2-10102-1
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)x2+1=0\frac{\cos{\left(x \right)}}{x^{2} + 1} = 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=−177.499984927823x_{2} = -177.499984927823
x3=73.8274273593601x_{3} = 73.8274273593601
x4=4.71238898038469x_{4} = 4.71238898038469
x5=39.2699081698724x_{5} = 39.2699081698724
x6=95.8185759344887x_{6} = 95.8185759344887
x7=45.553093477052x_{7} = 45.553093477052
x8=70.6858347057703x_{8} = 70.6858347057703
x9=−10.9955742875643x_{9} = -10.9955742875643
x10=−58.1194640914112x_{10} = -58.1194640914112
x11=−23.5619449019235x_{11} = -23.5619449019235
x12=26.7035375555132x_{12} = 26.7035375555132
x13=−26.7035375555132x_{13} = -26.7035375555132
x14=−89.5353906273091x_{14} = -89.5353906273091
x15=−17.2787595947439x_{15} = -17.2787595947439
x16=−42.4115008234622x_{16} = -42.4115008234622
x17=−61.261056745001x_{17} = -61.261056745001
x18=92.6769832808989x_{18} = 92.6769832808989
x19=−76.9690200129499x_{19} = -76.9690200129499
x20=−92.6769832808989x_{20} = -92.6769832808989
x21=−98.9601685880785x_{21} = -98.9601685880785
x22=61.261056745001x_{22} = 61.261056745001
x23=−54.9778714378214x_{23} = -54.9778714378214
x24=42.4115008234622x_{24} = 42.4115008234622
x25=−64.4026493985908x_{25} = -64.4026493985908
x26=67.5442420521806x_{26} = 67.5442420521806
x27=−7.85398163397448x_{27} = -7.85398163397448
x28=80.1106126665397x_{28} = 80.1106126665397
x29=−14.1371669411541x_{29} = -14.1371669411541
x30=14.1371669411541x_{30} = 14.1371669411541
x31=−1.5707963267949x_{31} = -1.5707963267949
x32=1.5707963267949x_{32} = 1.5707963267949
x33=108.384946548848x_{33} = 108.384946548848
x34=10.9955742875643x_{34} = 10.9955742875643
x35=17.2787595947439x_{35} = 17.2787595947439
x36=−51.8362787842316x_{36} = -51.8362787842316
x37=−29.845130209103x_{37} = -29.845130209103
x38=29.845130209103x_{38} = 29.845130209103
x39=−48.6946861306418x_{39} = -48.6946861306418
x40=−73.8274273593601x_{40} = -73.8274273593601
x41=23.5619449019235x_{41} = 23.5619449019235
x42=20.4203522483337x_{42} = 20.4203522483337
x43=−86.3937979737193x_{43} = -86.3937979737193
x44=54.9778714378214x_{44} = 54.9778714378214
x45=58.1194640914112x_{45} = 58.1194640914112
x46=51.8362787842316x_{46} = 51.8362787842316
x47=−67.5442420521806x_{47} = -67.5442420521806
x48=−4.71238898038469x_{48} = -4.71238898038469
x49=−45.553093477052x_{49} = -45.553093477052
x50=−70.6858347057703x_{50} = -70.6858347057703
x51=48.6946861306418x_{51} = 48.6946861306418
x52=−83.2522053201295x_{52} = -83.2522053201295
x53=−95.8185759344887x_{53} = -95.8185759344887
x54=89.5353906273091x_{54} = 89.5353906273091
x55=−39.2699081698724x_{55} = -39.2699081698724
x56=−306.305283725005x_{56} = -306.305283725005
x57=76.9690200129499x_{57} = 76.9690200129499
x58=−32.9867228626928x_{58} = -32.9867228626928
x59=−20.4203522483337x_{59} = -20.4203522483337
x60=−36.1283155162826x_{60} = -36.1283155162826
x61=7.85398163397448x_{61} = 7.85398163397448
x62=−80.1106126665397x_{62} = -80.1106126665397
x63=86.3937979737193x_{63} = 86.3937979737193
x64=98.9601685880785x_{64} = 98.9601685880785
x65=36.1283155162826x_{65} = 36.1283155162826
x66=64.4026493985908x_{66} = 64.4026493985908
x67=83.2522053201295x_{67} = 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^2 + 1).
cos⁡(0)02+1\frac{\cos{\left(0 \right)}}{0^{2} + 1}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 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
−2xcos⁡(x)(x2+1)2−sin⁡(x)x2+1=0- \frac{2 x \cos{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{\sin{\left(x \right)}}{x^{2} + 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=21.9002665401996x_{1} = 21.9002665401996
x2=75.3716994196716x_{2} = 75.3716994196716
x3=−87.9418588604656x_{3} = -87.9418588604656
x4=43.9368321750172x_{4} = 43.9368321750172
x5=12.4075674897868x_{5} = 12.4075674897868
x6=−75.3716994196716x_{6} = -75.3716994196716
x7=40.7917435749351x_{7} = 40.7917435749351
x8=−12.4075674897868x_{8} = -12.4075674897868
x9=−40.7917435749351x_{9} = -40.7917435749351
x10=−47.0814548776037x_{10} = -47.0814548776037
x11=−84.7994242303256x_{11} = -84.7994242303256
x12=−15.5808165061202x_{12} = -15.5808165061202
x13=25.0532062442974x_{13} = 25.0532062442974
x14=−113.079652107775x_{14} = -113.079652107775
x15=9.21343494397267x_{15} = 9.21343494397267
x16=−43.9368321750172x_{16} = -43.9368321750172
x17=−97.3688368618863x_{17} = -97.3688368618863
x18=−50.2256989863186x_{18} = -50.2256989863186
x19=15.5808165061202x_{19} = 15.5808165061202
x20=56.513303694752x_{20} = 56.513303694752
x21=−81.6569248421486x_{21} = -81.6569248421486
x22=34.4996609189666x_{22} = 34.4996609189666
x23=−103.653266658919x_{23} = -103.653266658919
x24=−37.6460727029451x_{24} = -37.6460727029451
x25=31.3522862210969x_{25} = 31.3522862210969
x26=72.2289536816917x_{26} = 72.2289536816917
x27=−100.511071203627x_{27} = -100.511071203627
x28=−106.7954266585x_{28} = -106.7954266585
x29=53.3696312768345x_{29} = 53.3696312768345
x30=28.203628119338x_{30} = 28.203628119338
x31=91.0842354305333x_{31} = 91.0842354305333
x32=62.8000247758447x_{32} = 62.8000247758447
x33=50.2256989863186x_{33} = 50.2256989863186
x34=−122.505790268738x_{34} = -122.505790268738
x35=−72.2289536816917x_{35} = -72.2289536816917
x36=0x_{36} = 0
x37=−5.96808139239822x_{37} = -5.96808139239822
x38=94.2265597456126x_{38} = 94.2265597456126
x39=−62.8000247758447x_{39} = -62.8000247758447
x40=81.6569248421486x_{40} = 81.6569248421486
x41=−65.9431328237524x_{41} = -65.9431328237524
x42=84.7994242303256x_{42} = 84.7994242303256
x43=78.5143529265667x_{43} = 78.5143529265667
x44=−34.4996609189666x_{44} = -34.4996609189666
x45=37.6460727029451x_{45} = 37.6460727029451
x46=−69.0861031389786x_{46} = -69.0861031389786
x47=−31.3522862210969x_{47} = -31.3522862210969
x48=163.350575451696x_{48} = 163.350575451696
x49=−91.0842354305333x_{49} = -91.0842354305333
x50=−94.2265597456126x_{50} = -94.2265597456126
x51=69.0861031389786x_{51} = 69.0861031389786
x52=−18.7435542483014x_{52} = -18.7435542483014
x53=−53.3696312768345x_{53} = -53.3696312768345
x54=2.54373214752609x_{54} = 2.54373214752609
x55=18.7435542483014x_{55} = 18.7435542483014
x56=59.656757255627x_{56} = 59.656757255627
x57=47.0814548776037x_{57} = 47.0814548776037
x58=65.9431328237524x_{58} = 65.9431328237524
x59=−25.0532062442974x_{59} = -25.0532062442974
x60=−59.656757255627x_{60} = -59.656757255627
x61=−78.5143529265667x_{61} = -78.5143529265667
x62=−56.513303694752x_{62} = -56.513303694752
x63=−21.9002665401996x_{63} = -21.9002665401996
x64=97.3688368618863x_{64} = 97.3688368618863
x65=−28.203628119338x_{65} = -28.203628119338
x66=100.511071203627x_{66} = 100.511071203627
x67=−9.21343494397267x_{67} = -9.21343494397267
x68=87.9418588604656x_{68} = 87.9418588604656
x69=−210.477206074369x_{69} = -210.477206074369
x70=−2.54373214752609x_{70} = -2.54373214752609
x71=5.96808139239822x_{71} = 5.96808139239822
Signos de extremos en los puntos:
(21.90026654019963, -0.00207205193264381)

(75.37169941967161, 0.00017593577340359)

(-87.94185886046559, 0.0001292529049009)

(43.936832175017194, 0.000517212000046997)

(12.40756748978677, 0.00637258289495849)

(-75.37169941967161, 0.00017593577340359)

(40.79174357493512, -0.000599892999132703)

(-12.40756748978677, 0.00637258289495849)

(-40.79174357493512, -0.000599892999132703)

(-47.08145487760369, -0.000450519125938963)

(-84.79942423032556, -0.00013900585084881)

(-15.580816506120234, -0.00406924940329345)

(25.053206244297428, 0.00158564848443144)

(-113.07965210777498, 7.81860375912636e-5)

(9.213434943972674, -0.011384094242491)

(-43.936832175017194, 0.000517212000046997)

(-97.3688368618863, -0.000105444189250915)

(-50.22569898631863, 0.000395942499274958)

(15.580816506120234, -0.00406924940329345)

(56.51330369475196, 0.000312817485971633)

(-81.6569248421486, 0.000149905871666022)

(34.49966091896661, -0.000838066136358659)

(-103.65326665891925, -9.30492289536518e-5)

(-37.64607270294512, 0.000704114293701762)

(31.352286221096882, 0.00101423808278872)

(72.22895368169175, -0.000191570111140939)

(-100.51107120362654, 9.89562584809543e-5)

(-106.79542665849998, 8.76557646972633e-5)

(53.36963127683454, -0.000350715231486932)

(28.203628119338006, -0.00125244284383629)

(91.08423543053327, -0.000120491545810595)

(62.80002477584475, 0.000253367116057383)

(50.22569898631863, 0.000395942499274958)

(-122.50579026873812, -6.66192853304118e-5)

(-72.22895368169175, -0.000191570111140939)

(0, 1)

(-5.968081392398221, 0.0259643971802455)

(94.22655974561256, 0.000112591766511704)

(-62.80002477584475, 0.000253367116057383)

(81.6569248421486, 0.000149905871666022)

(-65.94313282375245, -0.000229806033389755)

(84.79942423032556, -0.00013900585084881)

(78.51435292656672, -0.000162140173318783)

(-34.49966091896661, -0.000838066136358659)

(37.64607270294512, 0.000704114293701762)

(-69.0861031389786, 0.000209385109224912)

(-31.352286221096882, 0.00101423808278872)

(163.35057545169616, 3.74722559859326e-5)

(-91.08423543053327, -0.000120491545810595)

(-94.22655974561256, 0.000112591766511704)

(69.0861031389786, 0.000209385109224912)

(-18.7435542483014, 0.00282239086745388)

(-53.36963127683454, -0.000350715231486932)

(2.5437321475260917, -0.110639672191836)

(18.7435542483014, 0.00282239086745388)

(59.65675725562702, -0.000280746865913829)

(47.08145487760369, -0.000450519125938963)

(65.94313282375245, -0.000229806033389755)

(-25.053206244297428, 0.00158564848443144)

(-59.65675725562702, -0.000280746865913829)

(-78.51435292656672, -0.000162140173318783)

(-56.51330369475196, 0.000312817485971633)

(-21.90026654019963, -0.00207205193264381)

(97.3688368618863, -0.000105444189250915)

(-28.203628119338006, -0.00125244284383629)

(100.51107120362654, 9.89562584809543e-5)

(-9.213434943972674, -0.011384094242491)

(87.94185886046559, 0.0001292529049009)

(-210.47720607436906, -2.25715015693393e-5)

(-2.5437321475260917, -0.110639672191836)

(5.968081392398221, 0.0259643971802455)


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=21.9002665401996x_{1} = 21.9002665401996
x2=40.7917435749351x_{2} = 40.7917435749351
x3=−40.7917435749351x_{3} = -40.7917435749351
x4=−47.0814548776037x_{4} = -47.0814548776037
x5=−84.7994242303256x_{5} = -84.7994242303256
x6=−15.5808165061202x_{6} = -15.5808165061202
x7=9.21343494397267x_{7} = 9.21343494397267
x8=−97.3688368618863x_{8} = -97.3688368618863
x9=15.5808165061202x_{9} = 15.5808165061202
x10=34.4996609189666x_{10} = 34.4996609189666
x11=−103.653266658919x_{11} = -103.653266658919
x12=72.2289536816917x_{12} = 72.2289536816917
x13=53.3696312768345x_{13} = 53.3696312768345
x14=28.203628119338x_{14} = 28.203628119338
x15=91.0842354305333x_{15} = 91.0842354305333
x16=−122.505790268738x_{16} = -122.505790268738
x17=−72.2289536816917x_{17} = -72.2289536816917
x18=−65.9431328237524x_{18} = -65.9431328237524
x19=84.7994242303256x_{19} = 84.7994242303256
x20=78.5143529265667x_{20} = 78.5143529265667
x21=−34.4996609189666x_{21} = -34.4996609189666
x22=−91.0842354305333x_{22} = -91.0842354305333
x23=−53.3696312768345x_{23} = -53.3696312768345
x24=2.54373214752609x_{24} = 2.54373214752609
x25=59.656757255627x_{25} = 59.656757255627
x26=47.0814548776037x_{26} = 47.0814548776037
x27=65.9431328237524x_{27} = 65.9431328237524
x28=−59.656757255627x_{28} = -59.656757255627
x29=−78.5143529265667x_{29} = -78.5143529265667
x30=−21.9002665401996x_{30} = -21.9002665401996
x31=97.3688368618863x_{31} = 97.3688368618863
x32=−28.203628119338x_{32} = -28.203628119338
x33=−9.21343494397267x_{33} = -9.21343494397267
x34=−210.477206074369x_{34} = -210.477206074369
x35=−2.54373214752609x_{35} = -2.54373214752609
Puntos máximos de la función:
x35=75.3716994196716x_{35} = 75.3716994196716
x35=−87.9418588604656x_{35} = -87.9418588604656
x35=43.9368321750172x_{35} = 43.9368321750172
x35=12.4075674897868x_{35} = 12.4075674897868
x35=−75.3716994196716x_{35} = -75.3716994196716
x35=−12.4075674897868x_{35} = -12.4075674897868
x35=25.0532062442974x_{35} = 25.0532062442974
x35=−113.079652107775x_{35} = -113.079652107775
x35=−43.9368321750172x_{35} = -43.9368321750172
x35=−50.2256989863186x_{35} = -50.2256989863186
x35=56.513303694752x_{35} = 56.513303694752
x35=−81.6569248421486x_{35} = -81.6569248421486
x35=−37.6460727029451x_{35} = -37.6460727029451
x35=31.3522862210969x_{35} = 31.3522862210969
x35=−100.511071203627x_{35} = -100.511071203627
x35=−106.7954266585x_{35} = -106.7954266585
x35=62.8000247758447x_{35} = 62.8000247758447
x35=50.2256989863186x_{35} = 50.2256989863186
x35=0x_{35} = 0
x35=−5.96808139239822x_{35} = -5.96808139239822
x35=94.2265597456126x_{35} = 94.2265597456126
x35=−62.8000247758447x_{35} = -62.8000247758447
x35=81.6569248421486x_{35} = 81.6569248421486
x35=37.6460727029451x_{35} = 37.6460727029451
x35=−69.0861031389786x_{35} = -69.0861031389786
x35=−31.3522862210969x_{35} = -31.3522862210969
x35=163.350575451696x_{35} = 163.350575451696
x35=−94.2265597456126x_{35} = -94.2265597456126
x35=69.0861031389786x_{35} = 69.0861031389786
x35=−18.7435542483014x_{35} = -18.7435542483014
x35=18.7435542483014x_{35} = 18.7435542483014
x35=−25.0532062442974x_{35} = -25.0532062442974
x35=−56.513303694752x_{35} = -56.513303694752
x35=100.511071203627x_{35} = 100.511071203627
x35=87.9418588604656x_{35} = 87.9418588604656
x35=5.96808139239822x_{35} = 5.96808139239822
Decrece en los intervalos
[97.3688368618863,∞)\left[97.3688368618863, \infty\right)
Crece en los intervalos
(−∞,−210.477206074369]\left(-\infty, -210.477206074369\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
4xsin⁡(x)x2+1−cos⁡(x)+2(4x2x2+1−1)cos⁡(x)x2+1x2+1=0\frac{\frac{4 x \sin{\left(x \right)}}{x^{2} + 1} - \cos{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right) \cos{\left(x \right)}}{x^{2} + 1}}{x^{2} + 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=7.31236957667153x_{1} = 7.31236957667153
x2=−23.3910501690508x_{2} = -23.3910501690508
x3=−89.4906951115164x_{3} = -89.4906951115164
x4=−3.69928083952331x_{4} = -3.69928083952331
x5=−48.612414402203x_{5} = -48.612414402203
x6=−73.7732104776975x_{6} = -73.7732104776975
x7=−95.7768136964885x_{7} = -95.7768136964885
x8=86.3474755966677x_{8} = 86.3474755966677
x9=17.0443794915286x_{9} = 17.0443794915286
x10=−10.6203828275874x_{10} = -10.6203828275874
x11=−51.7590072918019x_{11} = -51.7590072918019
x12=−36.0172870463513x_{12} = -36.0172870463513
x13=−76.9170188649185x_{13} = -76.9170188649185
x14=−26.5529700363992x_{14} = -26.5529700363992
x15=−58.0505655345866x_{15} = -58.0505655345866
x16=−80.0606531586329x_{16} = -80.0606531586329
x17=−67.4849739703609x_{17} = -67.4849739703609
x18=3.69928083952331x_{18} = 3.69928083952331
x19=70.629204720493x_{19} = 70.629204720493
x20=0.5599347473979x_{20} = 0.5599347473979
x21=10.6203828275874x_{21} = 10.6203828275874
x22=463.376284124582x_{22} = 463.376284124582
x23=−64.3404851927513x_{23} = -64.3404851927513
x24=−29.7105507660002x_{24} = -29.7105507660002
x25=89.4906951115164x_{25} = 89.4906951115164
x26=83.2041331118179x_{26} = 83.2041331118179
x27=58.0505655345866x_{27} = 58.0505655345866
x28=64.3404851927513x_{28} = 64.3404851927513
x29=−13.8489274699853x_{29} = -13.8489274699853
x30=13.8489274699853x_{30} = 13.8489274699853
x31=−32.8650518133113x_{31} = -32.8650518133113
x32=39.1678061810769x_{32} = 39.1678061810769
x33=98.9197331449288x_{33} = 98.9197331449288
x34=−39.1678061810769x_{34} = -39.1678061810769
x35=−168.051405007597x_{35} = -168.051405007597
x36=80.0606531586329x_{36} = 80.0606531586329
x37=26.5529700363992x_{37} = 26.5529700363992
x38=−83.2041331118179x_{38} = -83.2041331118179
x39=36.0172870463513x_{39} = 36.0172870463513
x40=−86.3474755966677x_{40} = -86.3474755966677
x41=73.7732104776975x_{41} = 73.7732104776975
x42=−7.31236957667153x_{42} = -7.31236957667153
x43=−45.4651283904817x_{43} = -45.4651283904817
x44=51.7590072918019x_{44} = 51.7590072918019
x45=61.1956985466846x_{45} = 61.1956985466846
x46=−17.0443794915286x_{46} = -17.0443794915286
x47=54.9050265131646x_{47} = 54.9050265131646
x48=−92.6338041843149x_{48} = -92.6338041843149
x49=−20.2227299740537x_{49} = -20.2227299740537
x50=−70.629204720493x_{50} = -70.629204720493
x51=32.8650518133113x_{51} = 32.8650518133113
x52=67.4849739703609x_{52} = 67.4849739703609
x53=76.9170188649185x_{53} = 76.9170188649185
x54=20.2227299740537x_{54} = 20.2227299740537
x55=114.633238850285x_{55} = 114.633238850285
x56=−61.1956985466846x_{56} = -61.1956985466846
x57=92.6338041843149x_{57} = 92.6338041843149
x58=−42.316994092934x_{58} = -42.316994092934
x59=48.612414402203x_{59} = 48.612414402203
x60=−98.9197331449288x_{60} = -98.9197331449288
x61=149.198841587983x_{61} = 149.198841587983
x62=95.7768136964885x_{62} = 95.7768136964885
x63=45.4651283904817x_{63} = 45.4651283904817
x64=29.7105507660002x_{64} = 29.7105507660002
x65=23.3910501690508x_{65} = 23.3910501690508
x66=−54.9050265131646x_{66} = -54.9050265131646
x67=42.316994092934x_{67} = 42.316994092934

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
[114.633238850285,∞)\left[114.633238850285, \infty\right)
Convexa en los intervalos
(−∞,−168.051405007597]\left(-\infty, -168.051405007597\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(cos⁡(x)x2+1)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x^{2} + 1}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(cos⁡(x)x2+1)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x^{2} + 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)/(x^2 + 1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x)x(x2+1))=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x \left(x^{2} + 1\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x)x(x2+1))=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x \left(x^{2} + 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)x2+1=cos⁡(x)x2+1\frac{\cos{\left(x \right)}}{x^{2} + 1} = \frac{\cos{\left(x \right)}}{x^{2} + 1}
- Sí
cos⁡(x)x2+1=−cos⁡(x)x2+1\frac{\cos{\left(x \right)}}{x^{2} + 1} = - \frac{\cos{\left(x \right)}}{x^{2} + 1}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = cos(x)/(x^2+1)
    © Sr Examen – Калькулятор