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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x)
f(x) = ------
            2
       1 + x
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=−1.5707963267949x_{1} = -1.5707963267949
x2=−64.4026493985908x_{2} = -64.4026493985908
x3=76.9690200129499x_{3} = 76.9690200129499
x4=−23.5619449019235x_{4} = -23.5619449019235
x5=−58.1194640914112x_{5} = -58.1194640914112
x6=61.261056745001x_{6} = 61.261056745001
x7=80.1106126665397x_{7} = 80.1106126665397
x8=−48.6946861306418x_{8} = -48.6946861306418
x9=−29.845130209103x_{9} = -29.845130209103
x10=108.384946548848x_{10} = 108.384946548848
x11=−4.71238898038469x_{11} = -4.71238898038469
x12=−86.3937979737193x_{12} = -86.3937979737193
x13=−36.1283155162826x_{13} = -36.1283155162826
x14=−98.9601685880785x_{14} = -98.9601685880785
x15=1.5707963267949x_{15} = 1.5707963267949
x16=−39.2699081698724x_{16} = -39.2699081698724
x17=73.8274273593601x_{17} = 73.8274273593601
x18=−92.6769832808989x_{18} = -92.6769832808989
x19=42.4115008234622x_{19} = 42.4115008234622
x20=67.5442420521806x_{20} = 67.5442420521806
x21=−32.9867228626928x_{21} = -32.9867228626928
x22=14.1371669411541x_{22} = 14.1371669411541
x23=4.71238898038469x_{23} = 4.71238898038469
x24=32.9867228626928x_{24} = 32.9867228626928
x25=−177.499984927823x_{25} = -177.499984927823
x26=−10.9955742875643x_{26} = -10.9955742875643
x27=70.6858347057703x_{27} = 70.6858347057703
x28=36.1283155162826x_{28} = 36.1283155162826
x29=20.4203522483337x_{29} = 20.4203522483337
x30=−70.6858347057703x_{30} = -70.6858347057703
x31=−26.7035375555132x_{31} = -26.7035375555132
x32=10.9955742875643x_{32} = 10.9955742875643
x33=23.5619449019235x_{33} = 23.5619449019235
x34=45.553093477052x_{34} = 45.553093477052
x35=83.2522053201295x_{35} = 83.2522053201295
x36=−67.5442420521806x_{36} = -67.5442420521806
x37=−89.5353906273091x_{37} = -89.5353906273091
x38=−54.9778714378214x_{38} = -54.9778714378214
x39=95.8185759344887x_{39} = 95.8185759344887
x40=−17.2787595947439x_{40} = -17.2787595947439
x41=26.7035375555132x_{41} = 26.7035375555132
x42=17.2787595947439x_{42} = 17.2787595947439
x43=−42.4115008234622x_{43} = -42.4115008234622
x44=−306.305283725005x_{44} = -306.305283725005
x45=54.9778714378214x_{45} = 54.9778714378214
x46=−7.85398163397448x_{46} = -7.85398163397448
x47=48.6946861306418x_{47} = 48.6946861306418
x48=−51.8362787842316x_{48} = -51.8362787842316
x49=89.5353906273091x_{49} = 89.5353906273091
x50=92.6769832808989x_{50} = 92.6769832808989
x51=58.1194640914112x_{51} = 58.1194640914112
x52=−80.1106126665397x_{52} = -80.1106126665397
x53=−73.8274273593601x_{53} = -73.8274273593601
x54=86.3937979737193x_{54} = 86.3937979737193
x55=−76.9690200129499x_{55} = -76.9690200129499
x56=51.8362787842316x_{56} = 51.8362787842316
x57=39.2699081698724x_{57} = 39.2699081698724
x58=−20.4203522483337x_{58} = -20.4203522483337
x59=64.4026493985908x_{59} = 64.4026493985908
x60=−83.2522053201295x_{60} = -83.2522053201295
x61=98.9601685880785x_{61} = 98.9601685880785
x62=7.85398163397448x_{62} = 7.85398163397448
x63=−95.8185759344887x_{63} = -95.8185759344887
x64=−14.1371669411541x_{64} = -14.1371669411541
x65=29.845130209103x_{65} = 29.845130209103
x66=−45.553093477052x_{66} = -45.553093477052
x67=−61.261056745001x_{67} = -61.261056745001
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^2).
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=94.2265597456126x_{1} = 94.2265597456126
x2=100.511071203627x_{2} = 100.511071203627
x3=−43.9368321750172x_{3} = -43.9368321750172
x4=−69.0861031389786x_{4} = -69.0861031389786
x5=−34.4996609189666x_{5} = -34.4996609189666
x6=84.7994242303256x_{6} = 84.7994242303256
x7=69.0861031389786x_{7} = 69.0861031389786
x8=37.6460727029451x_{8} = 37.6460727029451
x9=−28.203628119338x_{9} = -28.203628119338
x10=−9.21343494397267x_{10} = -9.21343494397267
x11=−37.6460727029451x_{11} = -37.6460727029451
x12=−18.7435542483014x_{12} = -18.7435542483014
x13=9.21343494397267x_{13} = 9.21343494397267
x14=−84.7994242303256x_{14} = -84.7994242303256
x15=−50.2256989863186x_{15} = -50.2256989863186
x16=43.9368321750172x_{16} = 43.9368321750172
x17=−12.4075674897868x_{17} = -12.4075674897868
x18=−113.079652107775x_{18} = -113.079652107775
x19=−75.3716994196716x_{19} = -75.3716994196716
x20=−87.9418588604656x_{20} = -87.9418588604656
x21=53.3696312768345x_{21} = 53.3696312768345
x22=−97.3688368618863x_{22} = -97.3688368618863
x23=97.3688368618863x_{23} = 97.3688368618863
x24=2.54373214752609x_{24} = 2.54373214752609
x25=12.4075674897868x_{25} = 12.4075674897868
x26=18.7435542483014x_{26} = 18.7435542483014
x27=−72.2289536816917x_{27} = -72.2289536816917
x28=72.2289536816917x_{28} = 72.2289536816917
x29=5.96808139239822x_{29} = 5.96808139239822
x30=163.350575451696x_{30} = 163.350575451696
x31=0x_{31} = 0
x32=−103.653266658919x_{32} = -103.653266658919
x33=25.0532062442974x_{33} = 25.0532062442974
x34=−78.5143529265667x_{34} = -78.5143529265667
x35=−91.0842354305333x_{35} = -91.0842354305333
x36=31.3522862210969x_{36} = 31.3522862210969
x37=15.5808165061202x_{37} = 15.5808165061202
x38=28.203628119338x_{38} = 28.203628119338
x39=−106.7954266585x_{39} = -106.7954266585
x40=−53.3696312768345x_{40} = -53.3696312768345
x41=−81.6569248421486x_{41} = -81.6569248421486
x42=65.9431328237524x_{42} = 65.9431328237524
x43=−31.3522862210969x_{43} = -31.3522862210969
x44=34.4996609189666x_{44} = 34.4996609189666
x45=75.3716994196716x_{45} = 75.3716994196716
x46=−47.0814548776037x_{46} = -47.0814548776037
x47=59.656757255627x_{47} = 59.656757255627
x48=21.9002665401996x_{48} = 21.9002665401996
x49=−40.7917435749351x_{49} = -40.7917435749351
x50=−25.0532062442974x_{50} = -25.0532062442974
x51=−65.9431328237524x_{51} = -65.9431328237524
x52=78.5143529265667x_{52} = 78.5143529265667
x53=−210.477206074369x_{53} = -210.477206074369
x54=−15.5808165061202x_{54} = -15.5808165061202
x55=47.0814548776037x_{55} = 47.0814548776037
x56=−56.513303694752x_{56} = -56.513303694752
x57=−2.54373214752609x_{57} = -2.54373214752609
x58=50.2256989863186x_{58} = 50.2256989863186
x59=−94.2265597456126x_{59} = -94.2265597456126
x60=62.8000247758447x_{60} = 62.8000247758447
x61=−59.656757255627x_{61} = -59.656757255627
x62=−62.8000247758447x_{62} = -62.8000247758447
x63=91.0842354305333x_{63} = 91.0842354305333
x64=−100.511071203627x_{64} = -100.511071203627
x65=81.6569248421486x_{65} = 81.6569248421486
x66=56.513303694752x_{66} = 56.513303694752
x67=40.7917435749351x_{67} = 40.7917435749351
x68=−5.96808139239822x_{68} = -5.96808139239822
x69=−122.505790268738x_{69} = -122.505790268738
x70=87.9418588604656x_{70} = 87.9418588604656
x71=−21.9002665401996x_{71} = -21.9002665401996
Signos de extremos en los puntos:
(94.22655974561256, 0.000112591766511704)

(100.51107120362654, 9.89562584809543e-5)

(-43.936832175017194, 0.000517212000046997)

(-69.0861031389786, 0.000209385109224912)

(-34.49966091896661, -0.000838066136358659)

(84.79942423032556, -0.00013900585084881)

(69.0861031389786, 0.000209385109224912)

(37.64607270294512, 0.000704114293701762)

(-28.203628119338006, -0.00125244284383629)

(-9.213434943972674, -0.011384094242491)

(-37.64607270294512, 0.000704114293701762)

(-18.7435542483014, 0.00282239086745388)

(9.213434943972674, -0.011384094242491)

(-84.79942423032556, -0.00013900585084881)

(-50.22569898631863, 0.000395942499274958)

(43.936832175017194, 0.000517212000046997)

(-12.40756748978677, 0.00637258289495849)

(-113.07965210777498, 7.81860375912636e-5)

(-75.37169941967161, 0.00017593577340359)

(-87.94185886046559, 0.0001292529049009)

(53.36963127683454, -0.000350715231486932)

(-97.3688368618863, -0.000105444189250915)

(97.3688368618863, -0.000105444189250915)

(2.5437321475260917, -0.110639672191836)

(12.40756748978677, 0.00637258289495849)

(18.7435542483014, 0.00282239086745388)

(-72.22895368169175, -0.000191570111140939)

(72.22895368169175, -0.000191570111140939)

(5.968081392398221, 0.0259643971802455)

(163.35057545169616, 3.74722559859326e-5)

(0, 1)

(-103.65326665891925, -9.30492289536518e-5)

(25.053206244297428, 0.00158564848443144)

(-78.51435292656672, -0.000162140173318783)

(-91.08423543053327, -0.000120491545810595)

(31.352286221096882, 0.00101423808278872)

(15.580816506120234, -0.00406924940329345)

(28.203628119338006, -0.00125244284383629)

(-106.79542665849998, 8.76557646972633e-5)

(-53.36963127683454, -0.000350715231486932)

(-81.6569248421486, 0.000149905871666022)

(65.94313282375245, -0.000229806033389755)

(-31.352286221096882, 0.00101423808278872)

(34.49966091896661, -0.000838066136358659)

(75.37169941967161, 0.00017593577340359)

(-47.08145487760369, -0.000450519125938963)

(59.65675725562702, -0.000280746865913829)

(21.90026654019963, -0.00207205193264381)

(-40.79174357493512, -0.000599892999132703)

(-25.053206244297428, 0.00158564848443144)

(-65.94313282375245, -0.000229806033389755)

(78.51435292656672, -0.000162140173318783)

(-210.47720607436906, -2.25715015693393e-5)

(-15.580816506120234, -0.00406924940329345)

(47.08145487760369, -0.000450519125938963)

(-56.51330369475196, 0.000312817485971633)

(-2.5437321475260917, -0.110639672191836)

(50.22569898631863, 0.000395942499274958)

(-94.22655974561256, 0.000112591766511704)

(62.80002477584475, 0.000253367116057383)

(-59.65675725562702, -0.000280746865913829)

(-62.80002477584475, 0.000253367116057383)

(91.08423543053327, -0.000120491545810595)

(-100.51107120362654, 9.89562584809543e-5)

(81.6569248421486, 0.000149905871666022)

(56.51330369475196, 0.000312817485971633)

(40.79174357493512, -0.000599892999132703)

(-5.968081392398221, 0.0259643971802455)

(-122.50579026873812, -6.66192853304118e-5)

(87.94185886046559, 0.0001292529049009)

(-21.90026654019963, -0.00207205193264381)


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

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)/(1 + x^2), 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)/(1+x^2)
    © Sr Examen – Калькулятор