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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=−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=199.491133502952x_{6} = 199.491133502952
x7=61.261056745001x_{7} = 61.261056745001
x8=80.1106126665397x_{8} = 80.1106126665397
x9=−29.845130209103x_{9} = -29.845130209103
x10=−48.6946861306418x_{10} = -48.6946861306418
x11=1173.38485611579x_{11} = 1173.38485611579
x12=−4.71238898038469x_{12} = -4.71238898038469
x13=−86.3937979737193x_{13} = -86.3937979737193
x14=−36.1283155162826x_{14} = -36.1283155162826
x15=−98.9601685880785x_{15} = -98.9601685880785
x16=391.128285371929x_{16} = 391.128285371929
x17=−92.6769832808989x_{17} = -92.6769832808989
x18=−39.2699081698724x_{18} = -39.2699081698724
x19=73.8274273593601x_{19} = 73.8274273593601
x20=42.4115008234622x_{20} = 42.4115008234622
x21=67.5442420521806x_{21} = 67.5442420521806
x22=−32.9867228626928x_{22} = -32.9867228626928
x23=14.1371669411541x_{23} = 14.1371669411541
x24=4.71238898038469x_{24} = 4.71238898038469
x25=32.9867228626928x_{25} = 32.9867228626928
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=−714.712328691678x_{33} = -714.712328691678
x34=23.5619449019235x_{34} = 23.5619449019235
x35=45.553093477052x_{35} = 45.553093477052
x36=83.2522053201295x_{36} = 83.2522053201295
x37=−67.5442420521806x_{37} = -67.5442420521806
x38=−89.5353906273091x_{38} = -89.5353906273091
x39=−54.9778714378214x_{39} = -54.9778714378214
x40=95.8185759344887x_{40} = 95.8185759344887
x41=−17.2787595947439x_{41} = -17.2787595947439
x42=26.7035375555132x_{42} = 26.7035375555132
x43=17.2787595947439x_{43} = 17.2787595947439
x44=−42.4115008234622x_{44} = -42.4115008234622
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=−136.659280431156x_{60} = -136.659280431156
x61=−83.2522053201295x_{61} = -83.2522053201295
x62=538.78314009065x_{62} = 538.78314009065
x63=98.9601685880785x_{63} = 98.9601685880785
x64=7.85398163397448x_{64} = 7.85398163397448
x65=−95.8185759344887x_{65} = -95.8185759344887
x66=−14.1371669411541x_{66} = -14.1371669411541
x67=29.845130209103x_{67} = 29.845130209103
x68=−45.553093477052x_{68} = -45.553093477052
x69=−61.261056745001x_{69} = -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)/(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=−18.7990914357831x_{1} = -18.7990914357831
x2=−15.6479679638982x_{2} = -15.6479679638982
x3=15.6397620877646x_{3} = 15.6397620877646
x4=188.490225654589x_{4} = 188.490225654589
x5=100.520917114109x_{5} = 100.520917114109
x6=2.57625015820118x_{6} = 2.57625015820118
x7=−122.514017419839x_{7} = -122.514017419839
x8=56.5306616416093x_{8} = 56.5306616416093
x9=25.0912562079058x_{9} = 25.0912562079058
x10=6.08916120309943x_{10} = 6.08916120309943
x11=−72.242978694986x_{11} = -72.242978694986
x12=−546.635295693876x_{12} = -546.635295693876
x13=−37.673259943911x_{13} = -37.673259943911
x14=−91.0953290668266x_{14} = -91.0953290668266
x15=12.4794779911025x_{15} = 12.4794779911025
x16=91.0950880256329x_{16} = 91.0950880256329
x17=53.3879890840753x_{17} = 53.3879890840753
x18=75.3847808857452x_{18} = 75.3847808857452
x19=43.9590233567938x_{19} = 43.9590233567938
x20=−6.14411351301787x_{20} = -6.14411351301787
x21=−21.9475985837942x_{21} = -21.9475985837942
x22=−100.521115065812x_{22} = -100.521115065812
x23=40.8155939881502x_{23} = 40.8155939881502
x24=34.527701946778x_{24} = 34.527701946778
x25=−43.9600588531378x_{25} = -43.9600588531378
x26=−56.5312876685112x_{26} = -56.5312876685112
x27=191.631906205502x_{27} = 191.631906205502
x28=−65.9585122146304x_{28} = -65.9585122146304
x29=−40.8167952172419x_{29} = -40.8167952172419
x30=18.7934144113698x_{30} = 18.7934144113698
x31=−81.6693131963402x_{31} = -81.6693131963402
x32=37.6718497263809x_{32} = 37.6718497263809
x33=65.9580523911179x_{33} = 65.9580523911179
x34=28.2376364595748x_{34} = 28.2376364595748
x35=−59.6737803264459x_{35} = -59.6737803264459
x36=−94.2372799036618x_{36} = -94.2372799036618
x37=−50.2459712046114x_{37} = -50.2459712046114
x38=−69.1007741687956x_{38} = -69.1007741687956
x39=87.9530943542027x_{39} = 87.9530943542027
x40=−78.5272426949571x_{40} = -78.5272426949571
x41=−28.2401476526276x_{41} = -28.2401476526276
x42=−84.8113487041494x_{42} = -84.8113487041494
x43=−9.32825706323943x_{43} = -9.32825706323943
x44=78.5269183093816x_{44} = 78.5269183093816
x45=72.24259540785x_{45} = 72.24259540785
x46=50.2451786914948x_{46} = 50.2451786914948
x47=31.3830252979972x_{47} = 31.3830252979972
x48=9.30494468339504x_{48} = 9.30494468339504
x49=81.6690132946536x_{49} = 81.6690132946536
x50=47.1022022669651x_{50} = 47.1022022669651
x51=−1077.56535302402x_{51} = -1077.56535302402
x52=−97.379207861883x_{52} = -97.379207861883
x53=−2.88996969767843x_{53} = -2.88996969767843
x54=69.1003552230555x_{54} = 69.1003552230555
x55=−47.1031041186137x_{55} = -47.1031041186137
x56=−53.388691007263x_{56} = -53.388691007263
x57=21.9434371567881x_{57} = 21.9434371567881
x58=−62.8161843480611x_{58} = -62.8161843480611
x59=59.6732185170696x_{59} = 59.6732185170696
x60=97.378996929011x_{60} = 97.378996929011
x61=62.815677356778x_{61} = 62.815677356778
x62=−25.0944376288815x_{62} = -25.0944376288815
x63=94.2370546693974x_{63} = 94.2370546693974
x64=−12.492390025579x_{64} = -12.492390025579
x65=−75.3851328811964x_{65} = -75.3851328811964
x66=84.8110706151124x_{66} = 84.8110706151124
x67=−87.9533529268738x_{67} = -87.9533529268738
x68=−34.5293808983144x_{68} = -34.5293808983144
x69=−31.38505790634x_{69} = -31.38505790634
Signos de extremos en los puntos:
(-18.79909143578314, -0.0504430691319447)

(-15.647967963898166, 0.0599593189797558)

(15.63976208776456, -0.0681483206400774)

(188.4902256545889, 0.00533353551155559)

(100.52091711410945, 0.0100476316966419)

(2.5762501582011796, -0.535705052303484)

(-122.51401741983913, 0.00809598171844709)

(56.53066164160934, 0.0180051500304447)

(25.091256207905772, 0.0414731225016059)

(6.089161203099427, 0.192809042427521)

(-72.242978694986, 0.0136519134817116)

(-546.6352956938762, -0.00182602973305305)

(-37.673259943911006, -0.0258490197028825)

(-91.09532906682657, 0.0108576739325778)

(12.479477991102517, 0.0867833198945747)

(91.09508802563293, -0.0110987005999837)

(53.387989084075315, -0.0190848682073296)

(75.38478088574516, 0.0134423955413013)

(43.95902335679378, 0.0232716924030311)

(-6.1441135130178655, -0.138623930394573)

(-21.947598583794207, 0.0435362264748061)

(-100.52111506581193, -0.00984968979094353)

(40.81559398815024, -0.0251078697468112)

(34.52770194677802, -0.0298128246468963)

(-43.960058853137774, -0.022236464203186)

(-56.53128766851124, -0.0173792211238612)

(191.63190620550185, -0.00524563941809883)

(-65.95851221463039, 0.0149329557083856)

(-40.81679521724192, 0.023907001519389)

(18.793414411369753, 0.0561120230339157)

(-81.66931319634023, -0.0120955020439642)

(37.67184972638089, 0.0272587398500595)

(65.9580523911179, -0.0153927263543733)

(28.237636459574798, -0.0366891865463047)

(-59.67378032644585, 0.0164793457895915)

(-94.23727990366179, -0.0104995111118831)

(-50.24597120461141, -0.0195100148956696)

(-69.10077416879557, -0.0142637264671467)

(87.95309435420273, 0.0114996928375307)

(-78.52724269495707, 0.0125733134820883)

(-28.240147652627645, 0.0341795711715136)

(-84.81134870414938, 0.0116526790492257)

(-9.328257063239425, 0.0963710979823201)

(78.5269183093816, -0.0128976727485698)

(72.24259540785, -0.0140351638863266)

(50.245178691494786, 0.0203023709567303)

(31.38302529799723, 0.0328953023371544)

(9.304944683395044, -0.119546681963348)

(81.66901329465364, 0.0123953812433342)

(47.10220226696507, -0.0216858368023364)

(-1077.5653530240243, 0.000927157142018311)

(-97.37920786188297, 0.0101642243790071)

(-2.8899696976784344, 0.248976134877405)

(69.1003552230555, 0.0146826283229769)

(-47.10310411861372, 0.0207841885412821)

(-53.388691007263, 0.0183830682189117)

(21.94343715678808, -0.0476933188520339)

(-62.81618434806106, -0.0156680826074814)

(59.673218517069586, -0.0170410762454831)

(97.37899692901101, -0.0103751461271118)

(62.815677356778, 0.0161750096209984)

(-25.094437628881476, -0.0382942342355763)

(94.23705466939735, 0.010724732692878)

(-12.492390025578958, -0.0739131230459364)

(-75.38513288119637, -0.0130904310684593)

(84.81107061511238, -0.0119307487512748)

(-87.9533529268738, -0.0112411368826843)

(-34.5293808983144, 0.0281345781753277)

(-31.385057906339963, -0.0308637274812354)


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=−18.7990914357831x_{1} = -18.7990914357831
x2=15.6397620877646x_{2} = 15.6397620877646
x3=2.57625015820118x_{3} = 2.57625015820118
x4=−546.635295693876x_{4} = -546.635295693876
x5=−37.673259943911x_{5} = -37.673259943911
x6=91.0950880256329x_{6} = 91.0950880256329
x7=53.3879890840753x_{7} = 53.3879890840753
x8=−6.14411351301787x_{8} = -6.14411351301787
x9=−100.521115065812x_{9} = -100.521115065812
x10=40.8155939881502x_{10} = 40.8155939881502
x11=34.527701946778x_{11} = 34.527701946778
x12=−43.9600588531378x_{12} = -43.9600588531378
x13=−56.5312876685112x_{13} = -56.5312876685112
x14=191.631906205502x_{14} = 191.631906205502
x15=−81.6693131963402x_{15} = -81.6693131963402
x16=65.9580523911179x_{16} = 65.9580523911179
x17=28.2376364595748x_{17} = 28.2376364595748
x18=−94.2372799036618x_{18} = -94.2372799036618
x19=−50.2459712046114x_{19} = -50.2459712046114
x20=−69.1007741687956x_{20} = -69.1007741687956
x21=78.5269183093816x_{21} = 78.5269183093816
x22=72.24259540785x_{22} = 72.24259540785
x23=9.30494468339504x_{23} = 9.30494468339504
x24=47.1022022669651x_{24} = 47.1022022669651
x25=21.9434371567881x_{25} = 21.9434371567881
x26=−62.8161843480611x_{26} = -62.8161843480611
x27=59.6732185170696x_{27} = 59.6732185170696
x28=97.378996929011x_{28} = 97.378996929011
x29=−25.0944376288815x_{29} = -25.0944376288815
x30=−12.492390025579x_{30} = -12.492390025579
x31=−75.3851328811964x_{31} = -75.3851328811964
x32=84.8110706151124x_{32} = 84.8110706151124
x33=−87.9533529268738x_{33} = -87.9533529268738
x34=−31.38505790634x_{34} = -31.38505790634
Puntos máximos de la función:
x34=−15.6479679638982x_{34} = -15.6479679638982
x34=188.490225654589x_{34} = 188.490225654589
x34=100.520917114109x_{34} = 100.520917114109
x34=−122.514017419839x_{34} = -122.514017419839
x34=56.5306616416093x_{34} = 56.5306616416093
x34=25.0912562079058x_{34} = 25.0912562079058
x34=6.08916120309943x_{34} = 6.08916120309943
x34=−72.242978694986x_{34} = -72.242978694986
x34=−91.0953290668266x_{34} = -91.0953290668266
x34=12.4794779911025x_{34} = 12.4794779911025
x34=75.3847808857452x_{34} = 75.3847808857452
x34=43.9590233567938x_{34} = 43.9590233567938
x34=−21.9475985837942x_{34} = -21.9475985837942
x34=−65.9585122146304x_{34} = -65.9585122146304
x34=−40.8167952172419x_{34} = -40.8167952172419
x34=18.7934144113698x_{34} = 18.7934144113698
x34=37.6718497263809x_{34} = 37.6718497263809
x34=−59.6737803264459x_{34} = -59.6737803264459
x34=87.9530943542027x_{34} = 87.9530943542027
x34=−78.5272426949571x_{34} = -78.5272426949571
x34=−28.2401476526276x_{34} = -28.2401476526276
x34=−84.8113487041494x_{34} = -84.8113487041494
x34=−9.32825706323943x_{34} = -9.32825706323943
x34=50.2451786914948x_{34} = 50.2451786914948
x34=31.3830252979972x_{34} = 31.3830252979972
x34=81.6690132946536x_{34} = 81.6690132946536
x34=−1077.56535302402x_{34} = -1077.56535302402
x34=−97.379207861883x_{34} = -97.379207861883
x34=−2.88996969767843x_{34} = -2.88996969767843
x34=69.1003552230555x_{34} = 69.1003552230555
x34=−47.1031041186137x_{34} = -47.1031041186137
x34=−53.388691007263x_{34} = -53.388691007263
x34=62.815677356778x_{34} = 62.815677356778
x34=94.2370546693974x_{34} = 94.2370546693974
x34=−34.5293808983144x_{34} = -34.5293808983144
Decrece en los intervalos
[191.631906205502,∞)\left[191.631906205502, \infty\right)
Crece en los intervalos
(−∞,−546.635295693876]\left(-\infty, -546.635295693876\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=−61.2289118119026x_{1} = -61.2289118119026
x2=−51.7983897861238x_{2} = -51.7983897861238
x3=29.7755709323142x_{3} = 29.7755709323142
x4=−89.5132926274963x_{4} = -89.5132926274963
x5=83.2278802717944x_{5} = 83.2278802717944
x6=−83.228458145445x_{6} = -83.228458145445
x7=92.6551606286758x_{7} = 92.6551606286758
x8=−54.9421125829153x_{8} = -54.9421125829153
x9=−26.6310922236611x_{9} = -26.6310922236611
x10=−86.3709050594723x_{10} = -86.3709050594723
x11=98.9397464504172x_{11} = 98.9397464504172
x12=−39.22016138731x_{12} = -39.22016138731
x13=39.2175523279643x_{13} = 39.2175523279643
x14=32.9240332040206x_{14} = 32.9240332040206
x15=26.6254109350763x_{15} = 26.6254109350763
x16=67.5141687409854x_{16} = 67.5141687409854
x17=−168.063376807947x_{17} = -168.063376807947
x18=−42.3653647291314x_{18} = -42.3653647291314
x19=−48.654396838104x_{19} = -48.654396838104
x20=51.7968961320869x_{20} = 51.7968961320869
x21=89.5127931011103x_{21} = 89.5127931011103
x22=−10.825651157762x_{22} = -10.825651157762
x23=−73.80068645168x_{23} = -73.80068645168
x24=86.3703684986956x_{24} = 86.3703684986956
x25=45.5081427660817x_{25} = 45.5081427660817
x26=−92.6556268279389x_{26} = -92.6556268279389
x27=105.224163309626x_{27} = 105.224163309626
x28=58.0844210975337x_{28} = 58.0844210975337
x29=−4.32863617605124x_{29} = -4.32863617605124
x30=−70.657920700132x_{30} = -70.657920700132
x31=−237.181848301852x_{31} = -237.181848301852
x32=80.0853208283276x_{32} = 80.0853208283276
x33=−17.1684571899007x_{33} = -17.1684571899007
x34=−80.0859449790141x_{34} = -80.0859449790141
x35=10.7898786754269x_{35} = 10.7898786754269
x36=−105.224524739209x_{36} = -105.224524739209
x37=−67.5150472396589x_{37} = -67.5150472396589
x38=64.3710840254309x_{38} = 64.3710840254309
x39=70.6571186927646x_{39} = 70.6571186927646
x40=4.00507341668955x_{40} = 4.00507341668955
x41=−45.5100787997204x_{41} = -45.5100787997204
x42=54.9407852505616x_{42} = 54.9407852505616
x43=36.0712578833702x_{43} = 36.0712578833702
x44=7.54372449628009x_{44} = 7.54372449628009
x45=48.6527034788051x_{45} = 48.6527034788051
x46=−32.9277399444348x_{46} = -32.9277399444348
x47=−58.0856084395179x_{47} = -58.0856084395179
x48=−29.7801075137773x_{48} = -29.7801075137773
x49=23.4728313498836x_{49} = 23.4728313498836
x50=−36.0743437126941x_{50} = -36.0743437126941
x51=−76.9433575383977x_{51} = -76.9433575383977
x52=−98.9401552763972x_{52} = -98.9401552763972
x53=95.7974767616183x_{53} = 95.7974767616183
x54=567.053940733425x_{54} = 567.053940733425
x55=17.1546413657741x_{55} = 17.1546413657741
x56=20.3166301288662x_{56} = 20.3166301288662
x57=−20.3264348242219x_{57} = -20.3264348242219
x58=−23.4801553706306x_{58} = -23.4801553706306
x59=−7.61991323310644x_{59} = -7.61991323310644
x60=76.9426813176863x_{60} = 76.9426813176863
x61=61.2278434114583x_{61} = 61.2278434114583
x62=−64.3720505127272x_{62} = -64.3720505127272
x63=13.9825085391948x_{63} = 13.9825085391948
x64=287.448745694871x_{64} = 287.448745694871
x65=73.7999513585394x_{65} = 73.7999513585394
x66=42.3631297553676x_{66} = 42.3631297553676
x67=−95.797912862081x_{67} = -95.797912862081
x68=−14.0034717913284x_{68} = -14.0034717913284
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
[567.053940733425,∞)\left[567.053940733425, \infty\right)
Convexa en los intervalos
(−∞,−95.797912862081]\left(-\infty, -95.797912862081\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)−x−1\frac{\cos{\left(x \right)}}{x - 1} = \frac{\cos{\left(x \right)}}{- x - 1}
- No
cos⁡(x)x−1=−cos⁡(x)−x−1\frac{\cos{\left(x \right)}}{x - 1} = - \frac{\cos{\left(x \right)}}{- x - 1}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор