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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x)
f(x) = ------
       x + 1
f(x)=cos⁡(x)x+1f{\left(x \right)} = \frac{\cos{\left(x \right)}}{x + 1} 
f = cos(x)/(x + 1)
Gráfico de la función
02468-8-6-4-2-1010-2525
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)x+1=0\frac{\cos{\left(x \right)}}{x + 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=48.6946861306418x_{1} = 48.6946861306418
x2=92.6769832808989x_{2} = 92.6769832808989
x3=86.3937979737193x_{3} = 86.3937979737193
x4=−7.85398163397448x_{4} = -7.85398163397448
x5=−86.3937979737193x_{5} = -86.3937979737193
x6=1.5707963267949x_{6} = 1.5707963267949
x7=−64.4026493985908x_{7} = -64.4026493985908
x8=−58.1194640914112x_{8} = -58.1194640914112
x9=−83.2522053201295x_{9} = -83.2522053201295
x10=−161.792021659874x_{10} = -161.792021659874
x11=54.9778714378214x_{11} = 54.9778714378214
x12=89.5353906273091x_{12} = 89.5353906273091
x13=−54.9778714378214x_{13} = -54.9778714378214
x14=−20.4203522483337x_{14} = -20.4203522483337
x15=32.9867228626928x_{15} = 32.9867228626928
x16=−17.2787595947439x_{16} = -17.2787595947439
x17=23.5619449019235x_{17} = 23.5619449019235
x18=−256.039801267568x_{18} = -256.039801267568
x19=−45.553093477052x_{19} = -45.553093477052
x20=64.4026493985908x_{20} = 64.4026493985908
x21=45.553093477052x_{21} = 45.553093477052
x22=83.2522053201295x_{22} = 83.2522053201295
x23=−29.845130209103x_{23} = -29.845130209103
x24=−51.8362787842316x_{24} = -51.8362787842316
x25=230.90706003885x_{25} = 230.90706003885
x26=80.1106126665397x_{26} = 80.1106126665397
x27=−39.2699081698724x_{27} = -39.2699081698724
x28=−92.6769832808989x_{28} = -92.6769832808989
x29=4.71238898038469x_{29} = 4.71238898038469
x30=70.6858347057703x_{30} = 70.6858347057703
x31=36.1283155162826x_{31} = 36.1283155162826
x32=−70.6858347057703x_{32} = -70.6858347057703
x33=−48.6946861306418x_{33} = -48.6946861306418
x34=−127.234502470387x_{34} = -127.234502470387
x35=42.4115008234622x_{35} = 42.4115008234622
x36=−42.4115008234622x_{36} = -42.4115008234622
x37=−67.5442420521806x_{37} = -67.5442420521806
x38=10.9955742875643x_{38} = 10.9955742875643
x39=98.9601685880785x_{39} = 98.9601685880785
x40=−23.5619449019235x_{40} = -23.5619449019235
x41=20.4203522483337x_{41} = 20.4203522483337
x42=−61.261056745001x_{42} = -61.261056745001
x43=−10.9955742875643x_{43} = -10.9955742875643
x44=17.2787595947439x_{44} = 17.2787595947439
x45=−95.8185759344887x_{45} = -95.8185759344887
x46=−36.1283155162826x_{46} = -36.1283155162826
x47=61.261056745001x_{47} = 61.261056745001
x48=73.8274273593601x_{48} = 73.8274273593601
x49=14.1371669411541x_{49} = 14.1371669411541
x50=−26.7035375555132x_{50} = -26.7035375555132
x51=51.8362787842316x_{51} = 51.8362787842316
x52=−89.5353906273091x_{52} = -89.5353906273091
x53=39.2699081698724x_{53} = 39.2699081698724
x54=−32.9867228626928x_{54} = -32.9867228626928
x55=−14.1371669411541x_{55} = -14.1371669411541
x56=−4.71238898038469x_{56} = -4.71238898038469
x57=−76.9690200129499x_{57} = -76.9690200129499
x58=95.8185759344887x_{58} = 95.8185759344887
x59=76.9690200129499x_{59} = 76.9690200129499
x60=58.1194640914112x_{60} = 58.1194640914112
x61=−80.1106126665397x_{61} = -80.1106126665397
x62=−73.8274273593601x_{62} = -73.8274273593601
x63=7.85398163397448x_{63} = 7.85398163397448
x64=−1.5707963267949x_{64} = -1.5707963267949
x65=29.845130209103x_{65} = 29.845130209103
x66=67.5442420521806x_{66} = 67.5442420521806
x67=26.7035375555132x_{67} = 26.7035375555132
x68=−98.9601685880785x_{68} = -98.9601685880785
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 + 1).
cos⁡(0)1\frac{\cos{\left(0 \right)}}{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
−sin⁡(x)x+1−cos⁡(x)(x+1)2=0- \frac{\sin{\left(x \right)}}{x + 1} - \frac{\cos{\left(x \right)}}{\left(x + 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−34.527701946778x_{1} = -34.527701946778
x2=−31.3830252979972x_{2} = -31.3830252979972
x3=72.242978694986x_{3} = 72.242978694986
x4=−65.9580523911179x_{4} = -65.9580523911179
x5=18.7990914357831x_{5} = 18.7990914357831
x6=59.6737803264459x_{6} = 59.6737803264459
x7=−28.2376364595748x_{7} = -28.2376364595748
x8=−84.8110706151124x_{8} = -84.8110706151124
x9=−6.08916120309943x_{9} = -6.08916120309943
x10=−9.30494468339504x_{10} = -9.30494468339504
x11=−43.9590233567938x_{11} = -43.9590233567938
x12=15.6479679638982x_{12} = 15.6479679638982
x13=21.9475985837942x_{13} = 21.9475985837942
x14=172.781841669816x_{14} = 172.781841669816
x15=34.5293808983144x_{15} = 34.5293808983144
x16=6.14411351301787x_{16} = 6.14411351301787
x17=75.3851328811964x_{17} = 75.3851328811964
x18=2.88996969767843x_{18} = 2.88996969767843
x19=−78.5269183093816x_{19} = -78.5269183093816
x20=69.1007741687956x_{20} = 69.1007741687956
x21=−47.1022022669651x_{21} = -47.1022022669651
x22=−91.0950880256329x_{22} = -91.0950880256329
x23=56.5312876685112x_{23} = 56.5312876685112
x24=37.673259943911x_{24} = 37.673259943911
x25=50.2459712046114x_{25} = 50.2459712046114
x26=25.0944376288815x_{26} = 25.0944376288815
x27=91.0953290668266x_{27} = 91.0953290668266
x28=−37.6718497263809x_{28} = -37.6718497263809
x29=81.6693131963402x_{29} = 81.6693131963402
x30=47.1031041186137x_{30} = 47.1031041186137
x31=−87.9530943542027x_{31} = -87.9530943542027
x32=65.9585122146304x_{32} = 65.9585122146304
x33=−40.8155939881502x_{33} = -40.8155939881502
x34=−72.24259540785x_{34} = -72.24259540785
x35=−69.1003552230555x_{35} = -69.1003552230555
x36=−637.741738184573x_{36} = -637.741738184573
x37=−56.5306616416093x_{37} = -56.5306616416093
x38=−15.6397620877646x_{38} = -15.6397620877646
x39=43.9600588531378x_{39} = 43.9600588531378
x40=1313.18496827279x_{40} = 1313.18496827279
x41=94.2372799036618x_{41} = 94.2372799036618
x42=−75.3847808857452x_{42} = -75.3847808857452
x43=−94.2370546693974x_{43} = -94.2370546693974
x44=−100.520917114109x_{44} = -100.520917114109
x45=−62.815677356778x_{45} = -62.815677356778
x46=100.521115065812x_{46} = 100.521115065812
x47=40.8167952172419x_{47} = 40.8167952172419
x48=−81.6690132946536x_{48} = -81.6690132946536
x49=84.8113487041494x_{49} = 84.8113487041494
x50=−53.3879890840753x_{50} = -53.3879890840753
x51=−21.9434371567881x_{51} = -21.9434371567881
x52=−25.0912562079058x_{52} = -25.0912562079058
x53=12.492390025579x_{53} = 12.492390025579
x54=28.2401476526276x_{54} = 28.2401476526276
x55=197.915309953386x_{55} = 197.915309953386
x56=−97.378996929011x_{56} = -97.378996929011
x57=78.5272426949571x_{57} = 78.5272426949571
x58=−2.57625015820118x_{58} = -2.57625015820118
x59=62.8161843480611x_{59} = 62.8161843480611
x60=−59.6732185170696x_{60} = -59.6732185170696
x61=−12.4794779911025x_{61} = -12.4794779911025
x62=87.9533529268738x_{62} = 87.9533529268738
x63=31.38505790634x_{63} = 31.38505790634
x64=−18.7934144113698x_{64} = -18.7934144113698
x65=97.379207861883x_{65} = 97.379207861883
x66=9.32825706323943x_{66} = 9.32825706323943
x67=−50.2451786914948x_{67} = -50.2451786914948
x68=53.388691007263x_{68} = 53.388691007263
Signos de extremos en los puntos:
(-34.52770194677802, 0.0298128246468963)

(-31.38302529799723, -0.0328953023371544)

(72.242978694986, -0.0136519134817116)

(-65.9580523911179, 0.0153927263543733)

(18.79909143578314, 0.0504430691319447)

(59.67378032644585, -0.0164793457895915)

(-28.237636459574798, 0.0366891865463047)

(-84.81107061511238, 0.0119307487512748)

(-6.089161203099427, -0.192809042427521)

(-9.304944683395044, 0.119546681963348)

(-43.95902335679378, -0.0232716924030311)

(15.647967963898166, -0.0599593189797558)

(21.947598583794207, -0.0435362264748061)

(172.781841669816, -0.0057542458670116)

(34.5293808983144, -0.0281345781753277)

(6.1441135130178655, 0.138623930394573)

(75.38513288119637, 0.0130904310684593)

(2.8899696976784344, -0.248976134877405)

(-78.5269183093816, 0.0128976727485698)

(69.10077416879557, 0.0142637264671467)

(-47.10220226696507, 0.0216858368023364)

(-91.09508802563293, 0.0110987005999837)

(56.53128766851124, 0.0173792211238612)

(37.673259943911006, 0.0258490197028825)

(50.24597120461141, 0.0195100148956696)

(25.094437628881476, 0.0382942342355763)

(91.09532906682657, -0.0108576739325778)

(-37.67184972638089, -0.0272587398500595)

(81.66931319634023, 0.0120955020439642)

(47.10310411861372, -0.0207841885412821)

(-87.95309435420273, -0.0114996928375307)

(65.95851221463039, -0.0149329557083856)

(-40.81559398815024, 0.0251078697468112)

(-72.24259540785, 0.0140351638863266)

(-69.1003552230555, -0.0146826283229769)

(-637.7417381845734, 0.00157049350907233)

(-56.53066164160934, -0.0180051500304447)

(-15.63976208776456, 0.0681483206400774)

(43.960058853137774, 0.022236464203186)

(1313.1849682727866, 0.000760927673528925)

(94.23727990366179, 0.0104995111118831)

(-75.38478088574516, -0.0134423955413013)

(-94.23705466939735, -0.010724732692878)

(-100.52091711410945, -0.0100476316966419)

(-62.815677356778, -0.0161750096209984)

(100.52111506581193, 0.00984968979094353)

(40.81679521724192, -0.023907001519389)

(-81.66901329465364, -0.0123953812433342)

(84.81134870414938, -0.0116526790492257)

(-53.387989084075315, 0.0190848682073296)

(-21.94343715678808, 0.0476933188520339)

(-25.091256207905772, -0.0414731225016059)

(12.492390025578958, 0.0739131230459364)

(28.240147652627645, -0.0341795711715136)

(197.91530995338616, -0.00502720159537522)

(-97.37899692901101, 0.0103751461271118)

(78.52724269495707, -0.0125733134820883)

(-2.5762501582011796, 0.535705052303484)

(62.81618434806106, 0.0156680826074814)

(-59.673218517069586, 0.0170410762454831)

(-12.479477991102517, -0.0867833198945747)

(87.9533529268738, 0.0112411368826843)

(31.385057906339963, 0.0308637274812354)

(-18.793414411369753, -0.0561120230339157)

(97.37920786188297, -0.0101642243790071)

(9.328257063239425, -0.0963710979823201)

(-50.245178691494786, -0.0203023709567303)

(53.388691007263, -0.0183830682189117)


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=−31.3830252979972x_{1} = -31.3830252979972
x2=72.242978694986x_{2} = 72.242978694986
x3=59.6737803264459x_{3} = 59.6737803264459
x4=−6.08916120309943x_{4} = -6.08916120309943
x5=−43.9590233567938x_{5} = -43.9590233567938
x6=15.6479679638982x_{6} = 15.6479679638982
x7=21.9475985837942x_{7} = 21.9475985837942
x8=172.781841669816x_{8} = 172.781841669816
x9=34.5293808983144x_{9} = 34.5293808983144
x10=2.88996969767843x_{10} = 2.88996969767843
x11=91.0953290668266x_{11} = 91.0953290668266
x12=−37.6718497263809x_{12} = -37.6718497263809
x13=47.1031041186137x_{13} = 47.1031041186137
x14=−87.9530943542027x_{14} = -87.9530943542027
x15=65.9585122146304x_{15} = 65.9585122146304
x16=−69.1003552230555x_{16} = -69.1003552230555
x17=−56.5306616416093x_{17} = -56.5306616416093
x18=−75.3847808857452x_{18} = -75.3847808857452
x19=−94.2370546693974x_{19} = -94.2370546693974
x20=−100.520917114109x_{20} = -100.520917114109
x21=−62.815677356778x_{21} = -62.815677356778
x22=40.8167952172419x_{22} = 40.8167952172419
x23=−81.6690132946536x_{23} = -81.6690132946536
x24=84.8113487041494x_{24} = 84.8113487041494
x25=−25.0912562079058x_{25} = -25.0912562079058
x26=28.2401476526276x_{26} = 28.2401476526276
x27=197.915309953386x_{27} = 197.915309953386
x28=78.5272426949571x_{28} = 78.5272426949571
x29=−12.4794779911025x_{29} = -12.4794779911025
x30=−18.7934144113698x_{30} = -18.7934144113698
x31=97.379207861883x_{31} = 97.379207861883
x32=9.32825706323943x_{32} = 9.32825706323943
x33=−50.2451786914948x_{33} = -50.2451786914948
x34=53.388691007263x_{34} = 53.388691007263
Puntos máximos de la función:
x34=−34.527701946778x_{34} = -34.527701946778
x34=−65.9580523911179x_{34} = -65.9580523911179
x34=18.7990914357831x_{34} = 18.7990914357831
x34=−28.2376364595748x_{34} = -28.2376364595748
x34=−84.8110706151124x_{34} = -84.8110706151124
x34=−9.30494468339504x_{34} = -9.30494468339504
x34=6.14411351301787x_{34} = 6.14411351301787
x34=75.3851328811964x_{34} = 75.3851328811964
x34=−78.5269183093816x_{34} = -78.5269183093816
x34=69.1007741687956x_{34} = 69.1007741687956
x34=−47.1022022669651x_{34} = -47.1022022669651
x34=−91.0950880256329x_{34} = -91.0950880256329
x34=56.5312876685112x_{34} = 56.5312876685112
x34=37.673259943911x_{34} = 37.673259943911
x34=50.2459712046114x_{34} = 50.2459712046114
x34=25.0944376288815x_{34} = 25.0944376288815
x34=81.6693131963402x_{34} = 81.6693131963402
x34=−40.8155939881502x_{34} = -40.8155939881502
x34=−72.24259540785x_{34} = -72.24259540785
x34=−637.741738184573x_{34} = -637.741738184573
x34=−15.6397620877646x_{34} = -15.6397620877646
x34=43.9600588531378x_{34} = 43.9600588531378
x34=1313.18496827279x_{34} = 1313.18496827279
x34=94.2372799036618x_{34} = 94.2372799036618
x34=100.521115065812x_{34} = 100.521115065812
x34=−53.3879890840753x_{34} = -53.3879890840753
x34=−21.9434371567881x_{34} = -21.9434371567881
x34=12.492390025579x_{34} = 12.492390025579
x34=−97.378996929011x_{34} = -97.378996929011
x34=−2.57625015820118x_{34} = -2.57625015820118
x34=62.8161843480611x_{34} = 62.8161843480611
x34=−59.6732185170696x_{34} = -59.6732185170696
x34=87.9533529268738x_{34} = 87.9533529268738
x34=31.38505790634x_{34} = 31.38505790634
Decrece en los intervalos
[197.915309953386,∞)\left[197.915309953386, \infty\right)
Crece en los intervalos
(−∞,−100.520917114109]\left(-\infty, -100.520917114109\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+1+2cos⁡(x)(x+1)2x+1=0\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=80.0859449790141x_{1} = 80.0859449790141
x2=29.7801075137773x_{2} = 29.7801075137773
x3=86.3709050594723x_{3} = 86.3709050594723
x4=73.80068645168x_{4} = 73.80068645168
x5=76.9433575383977x_{5} = 76.9433575383977
x6=36.0743437126941x_{6} = 36.0743437126941
x7=20.3264348242219x_{7} = 20.3264348242219
x8=−29.7755709323142x_{8} = -29.7755709323142
x9=−48.6527034788051x_{9} = -48.6527034788051
x10=98.9401552763972x_{10} = 98.9401552763972
x11=136.64474976163x_{11} = 136.64474976163
x12=−64.3710840254309x_{12} = -64.3710840254309
x13=102.082358062481x_{13} = 102.082358062481
x14=95.797912862081x_{14} = 95.797912862081
x15=45.5100787997204x_{15} = 45.5100787997204
x16=−83.2278802717944x_{16} = -83.2278802717944
x17=−26.6254109350763x_{17} = -26.6254109350763
x18=58.0856084395179x_{18} = 58.0856084395179
x19=−89.5127931011103x_{19} = -89.5127931011103
x20=−80.0853208283276x_{20} = -80.0853208283276
x21=−13.9825085391948x_{21} = -13.9825085391948
x22=−45.5081427660817x_{22} = -45.5081427660817
x23=−7.54372449628009x_{23} = -7.54372449628009
x24=−39.2175523279643x_{24} = -39.2175523279643
x25=23.4801553706306x_{25} = 23.4801553706306
x26=32.9277399444348x_{26} = 32.9277399444348
x27=−23.4728313498836x_{27} = -23.4728313498836
x28=61.2289118119026x_{28} = 61.2289118119026
x29=−54.9407852505616x_{29} = -54.9407852505616
x30=64.3720505127272x_{30} = 64.3720505127272
x31=10.825651157762x_{31} = 10.825651157762
x32=−92.6551606286758x_{32} = -92.6551606286758
x33=−42.3631297553676x_{33} = -42.3631297553676
x34=89.5132926274963x_{34} = 89.5132926274963
x35=4.32863617605124x_{35} = 4.32863617605124
x36=14.0034717913284x_{36} = 14.0034717913284
x37=42.3653647291314x_{37} = 42.3653647291314
x38=51.7983897861238x_{38} = 51.7983897861238
x39=−76.9426813176863x_{39} = -76.9426813176863
x40=−32.9240332040206x_{40} = -32.9240332040206
x41=17.1684571899007x_{41} = 17.1684571899007
x42=67.5150472396589x_{42} = 67.5150472396589
x43=−20.3166301288662x_{43} = -20.3166301288662
x44=26.6310922236611x_{44} = 26.6310922236611
x45=−67.5141687409854x_{45} = -67.5141687409854
x46=−61.2278434114583x_{46} = -61.2278434114583
x47=−36.0712578833702x_{47} = -36.0712578833702
x48=−4.00507341668955x_{48} = -4.00507341668955
x49=54.9421125829153x_{49} = 54.9421125829153
x50=−17.1546413657741x_{50} = -17.1546413657741
x51=−98.9397464504172x_{51} = -98.9397464504172
x52=48.654396838104x_{52} = 48.654396838104
x53=−70.6571186927646x_{53} = -70.6571186927646
x54=−95.7974767616183x_{54} = -95.7974767616183
x55=−86.3703684986956x_{55} = -86.3703684986956
x56=39.22016138731x_{56} = 39.22016138731
x57=7.61991323310644x_{57} = 7.61991323310644
x58=−58.0844210975337x_{58} = -58.0844210975337
x59=83.228458145445x_{59} = 83.228458145445
x60=70.657920700132x_{60} = 70.657920700132
x61=92.6556268279389x_{61} = 92.6556268279389
x62=−10.7898786754269x_{62} = -10.7898786754269
x63=−51.7968961320869x_{63} = -51.7968961320869
x64=−73.7999513585394x_{64} = -73.7999513585394
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)+2sin⁡(x)x+1+2cos⁡(x)(x+1)2x+1)=−∞\lim_{x \to -1^-}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1}\right) = -\infty
lim⁡x→−1+(−cos⁡(x)+2sin⁡(x)x+1+2cos⁡(x)(x+1)2x+1)=∞\lim_{x \to -1^+}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \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
[102.082358062481,∞)\left[102.082358062481, \infty\right)
Convexa en los intervalos
(−∞,−95.7974767616183]\left(-\infty, -95.7974767616183\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)x+1)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \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)x+1)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \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)/(x + 1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x)x(x+1))=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \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)x(x+1))=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \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)x+1=cos⁡(x)1−x\frac{\cos{\left(x \right)}}{x + 1} = \frac{\cos{\left(x \right)}}{1 - x}
- No
cos⁡(x)x+1=−cos⁡(x)1−x\frac{\cos{\left(x \right)}}{x + 1} = - \frac{\cos{\left(x \right)}}{1 - x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор