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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)1−x2f{\left(x \right)} = \frac{\cos{\left(x \right)}}{1 - x^{2}} 
f = cos(x)/(1 - x^2)
Gráfico de la función
02468-8-6-4-2-1010-1010
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−1x_{1} = -1
x2=1x_{2} = 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)1−x2=0\frac{\cos{\left(x \right)}}{1 - 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=−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=−92.6769832808989x_{15} = -92.6769832808989
x16=−39.2699081698724x_{16} = -39.2699081698724
x17=73.8274273593601x_{17} = 73.8274273593601
x18=42.4115008234622x_{18} = 42.4115008234622
x19=67.5442420521806x_{19} = 67.5442420521806
x20=−32.9867228626928x_{20} = -32.9867228626928
x21=14.1371669411541x_{21} = 14.1371669411541
x22=−523.075176822701x_{22} = -523.075176822701
x23=4.71238898038469x_{23} = 4.71238898038469
x24=32.9867228626928x_{24} = 32.9867228626928
x25=−10.9955742875643x_{25} = -10.9955742875643
x26=70.6858347057703x_{26} = 70.6858347057703
x27=36.1283155162826x_{27} = 36.1283155162826
x28=−105.243353895258x_{28} = -105.243353895258
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=702.145958077319x_{44} = 702.145958077319
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)1−02\frac{\cos{\left(0 \right)}}{1 - 0^{2}}
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)(1−x2)2−sin⁡(x)1−x2=0\frac{2 x \cos{\left(x \right)}}{\left(1 - x^{2}\right)^{2}} - \frac{\sin{\left(x \right)}}{1 - x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−53.3696049818501x_{1} = -53.3696049818501
x2=31.3521566903887x_{2} = 31.3521566903887
x3=78.5143446648172x_{3} = 78.5143446648172
x4=−15.5797675022891x_{4} = -15.5797675022891
x5=97.3688325296866x_{5} = 97.3688325296866
x6=−21.8998872970823x_{6} = -21.8998872970823
x7=−97.3688325296866x_{7} = -97.3688325296866
x8=−87.941852980689x_{8} = -87.941852980689
x9=−56.5132815466599x_{9} = -56.5132815466599
x10=59.656738426191x_{10} = 59.656738426191
x11=81.6569174978428x_{11} = 81.6569174978428
x12=50.2256674407532x_{12} = 50.2256674407532
x13=−12.4054996335861x_{13} = -12.4054996335861
x14=−100.511067265113x_{14} = -100.511067265113
x15=28.2034502671317x_{15} = 28.2034502671317
x16=37.6459978360151x_{16} = 37.6459978360151
x17=−34.4995636692158x_{17} = -34.4995636692158
x18=72.2289430706097x_{18} = 72.2289430706097
x19=2.33112237041442x_{19} = 2.33112237041442
x20=−50.2256674407532x_{20} = -50.2256674407532
x21=84.7994176724893x_{21} = 84.7994176724893
x22=−78.5143446648172x_{22} = -78.5143446648172
x23=125.647788969162x_{23} = 125.647788969162
x24=34.4995636692158x_{24} = 34.4995636692158
x25=−37.6459978360151x_{25} = -37.6459978360151
x26=25.0529526753384x_{26} = 25.0529526753384
x27=−9.20843355440115x_{27} = -9.20843355440115
x28=9.20843355440115x_{28} = 9.20843355440115
x29=−47.0814165846103x_{29} = -47.0814165846103
x30=53.3696049818501x_{30} = 53.3696049818501
x31=100.511067265113x_{31} = 100.511067265113
x32=91.0842301384618x_{32} = 91.0842301384618
x33=18.7429502117119x_{33} = 18.7429502117119
x34=0x_{34} = 0
x35=−18.7429502117119x_{35} = -18.7429502117119
x36=40.7916847146183x_{36} = 40.7916847146183
x37=−69.086091013299x_{37} = -69.086091013299
x38=62.8000086337252x_{38} = 62.8000086337252
x39=−103.653263067797x_{39} = -103.653263067797
x40=65.943118880897x_{40} = 65.943118880897
x41=131.93173238582x_{41} = 131.93173238582
x42=−62.8000086337252x_{42} = -62.8000086337252
x43=21.8998872970823x_{43} = 21.8998872970823
x44=5.95017264337656x_{44} = 5.95017264337656
x45=−31.3521566903887x_{45} = -31.3521566903887
x46=−94.2265549654551x_{46} = -94.2265549654551
x47=−40.7916847146183x_{47} = -40.7916847146183
x48=12.4054996335861x_{48} = 12.4054996335861
x49=−75.3716900810604x_{49} = -75.3716900810604
x50=47.0814165846103x_{50} = 47.0814165846103
x51=56.5132815466599x_{51} = 56.5132815466599
x52=−43.9367850637406x_{52} = -43.9367850637406
x53=−59.656738426191x_{53} = -59.656738426191
x54=−25.0529526753384x_{54} = -25.0529526753384
x55=94.2265549654551x_{55} = 94.2265549654551
x56=−84.7994176724893x_{56} = -84.7994176724893
x57=−2.33112237041442x_{57} = -2.33112237041442
x58=15.5797675022891x_{58} = 15.5797675022891
x59=43.9367850637406x_{59} = 43.9367850637406
x60=−72.2289430706097x_{60} = -72.2289430706097
x61=−91.0842301384618x_{61} = -91.0842301384618
x62=75.3716900810604x_{62} = 75.3716900810604
x63=69.086091013299x_{63} = 69.086091013299
x64=−5.95017264337656x_{64} = -5.95017264337656
x65=87.941852980689x_{65} = 87.941852980689
x66=−65.943118880897x_{66} = -65.943118880897
x67=−28.2034502671317x_{67} = -28.2034502671317
x68=−81.6569174978428x_{68} = -81.6569174978428
Signos de extremos en los puntos:
(-53.36960498185014, 0.000350961579426066)

(31.352156690388735, -0.0010163038211767)

(78.51434466481717, 0.000162192786313112)

(-15.579767502289146, 0.00410291496827567)

(97.36883252968656, 0.000105466435587997)

(-21.89988729708232, 0.00208071049534438)

(-97.36883252968656, 0.000105466435587997)

(-87.94185298068903, -0.000129286334811403)

(-56.51328154665989, -0.000313013440723007)

(59.65673842619101, 0.000280904680743359)

(81.6569174978428, -0.000149950842172114)

(50.22566744075319, -0.000396256537564922)

(-12.405499633586086, -0.00645592708029144)

(-100.51106726511297, -9.89758509183166e-5)

(28.203450267131746, 0.00125559586593523)

(37.645997836015106, -0.000705108648137029)

(-34.49956366921579, 0.000839475570857111)

(72.2289430706097, 0.000191643565642655)

(2.331122370414423, 0.155421131677418)

(-50.22566744075319, -0.000396256537564922)

(84.79941767248933, 0.00013904451760157)

(-78.51434466481717, 0.000162192786313112)

(125.64778896916187, -6.33377731879486e-5)

(34.49956366921579, 0.000839475570857111)

(-37.645997836015106, -0.000705108648137029)

(25.0529526753384, -0.0015907091444567)

(-9.208433554401154, 0.0116556571276676)

(9.208433554401154, 0.0116556571276676)

(-47.0814165846103, 0.000450925793751296)

(53.36960498185014, 0.000350961579426066)

(100.51106726511297, -9.89758509183166e-5)

(91.0842301384618, 0.000120520596237142)

(18.742950211711907, -0.00283850456596173)

(0, 1)

(-18.742950211711907, -0.00283850456596173)

(40.791684714618334, 0.000600614473584034)

(-69.08609101329898, -0.000209472866973509)

(62.80000863372525, -0.000253495636099399)

(-103.65326306779691, 9.30665517191495e-5)

(65.94311888089696, 0.000229911752195002)

(131.93173238582048, -5.74482124130948e-5)

(-62.80000863372525, -0.000253495636099399)

(21.89988729708232, 0.00208071049534438)

(5.9501726433765585, -0.0274690904508921)

(-31.352156690388735, -0.0010163038211767)

(-94.22655496545507, -0.000112617131747143)

(-40.791684714618334, 0.000600614473584034)

(12.405499633586086, -0.00645592708029144)

(-75.37169008106044, -0.000175997723793477)

(47.0814165846103, 0.000450925793751296)

(56.51328154665989, -0.000313013440723007)

(-43.936785063740594, -0.000517748125715798)

(-59.65673842619101, 0.000280904680743359)

(-25.0529526753384, -0.0015907091444567)

(94.22655496545507, -0.000112617131747143)

(-84.79941767248933, 0.00013904451760157)

(-2.331122370414423, 0.155421131677418)

(15.579767502289146, 0.00410291496827567)

(43.936785063740594, -0.000517748125715798)

(-72.2289430706097, 0.000191643565642655)

(-91.0842301384618, 0.000120520596237142)

(75.37169008106044, -0.000175997723793477)

(69.08609101329898, -0.000209472866973509)

(-5.9501726433765585, -0.0274690904508921)

(87.94185298068903, -0.000129286334811403)

(-65.94311888089696, 0.000229911752195002)

(-28.203450267131746, 0.00125559586593523)

(-81.6569174978428, -0.000149950842172114)


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=31.3521566903887x_{1} = 31.3521566903887
x2=−87.941852980689x_{2} = -87.941852980689
x3=−56.5132815466599x_{3} = -56.5132815466599
x4=81.6569174978428x_{4} = 81.6569174978428
x5=50.2256674407532x_{5} = 50.2256674407532
x6=−12.4054996335861x_{6} = -12.4054996335861
x7=−100.511067265113x_{7} = -100.511067265113
x8=37.6459978360151x_{8} = 37.6459978360151
x9=−50.2256674407532x_{9} = -50.2256674407532
x10=125.647788969162x_{10} = 125.647788969162
x11=−37.6459978360151x_{11} = -37.6459978360151
x12=25.0529526753384x_{12} = 25.0529526753384
x13=100.511067265113x_{13} = 100.511067265113
x14=18.7429502117119x_{14} = 18.7429502117119
x15=0x_{15} = 0
x16=−18.7429502117119x_{16} = -18.7429502117119
x17=−69.086091013299x_{17} = -69.086091013299
x18=62.8000086337252x_{18} = 62.8000086337252
x19=131.93173238582x_{19} = 131.93173238582
x20=−62.8000086337252x_{20} = -62.8000086337252
x21=5.95017264337656x_{21} = 5.95017264337656
x22=−31.3521566903887x_{22} = -31.3521566903887
x23=−94.2265549654551x_{23} = -94.2265549654551
x24=12.4054996335861x_{24} = 12.4054996335861
x25=−75.3716900810604x_{25} = -75.3716900810604
x26=56.5132815466599x_{26} = 56.5132815466599
x27=−43.9367850637406x_{27} = -43.9367850637406
x28=−25.0529526753384x_{28} = -25.0529526753384
x29=94.2265549654551x_{29} = 94.2265549654551
x30=43.9367850637406x_{30} = 43.9367850637406
x31=75.3716900810604x_{31} = 75.3716900810604
x32=69.086091013299x_{32} = 69.086091013299
x33=−5.95017264337656x_{33} = -5.95017264337656
x34=87.941852980689x_{34} = 87.941852980689
x35=−81.6569174978428x_{35} = -81.6569174978428
Puntos máximos de la función:
x35=−53.3696049818501x_{35} = -53.3696049818501
x35=78.5143446648172x_{35} = 78.5143446648172
x35=−15.5797675022891x_{35} = -15.5797675022891
x35=97.3688325296866x_{35} = 97.3688325296866
x35=−21.8998872970823x_{35} = -21.8998872970823
x35=−97.3688325296866x_{35} = -97.3688325296866
x35=59.656738426191x_{35} = 59.656738426191
x35=28.2034502671317x_{35} = 28.2034502671317
x35=−34.4995636692158x_{35} = -34.4995636692158
x35=72.2289430706097x_{35} = 72.2289430706097
x35=2.33112237041442x_{35} = 2.33112237041442
x35=84.7994176724893x_{35} = 84.7994176724893
x35=−78.5143446648172x_{35} = -78.5143446648172
x35=34.4995636692158x_{35} = 34.4995636692158
x35=−9.20843355440115x_{35} = -9.20843355440115
x35=9.20843355440115x_{35} = 9.20843355440115
x35=−47.0814165846103x_{35} = -47.0814165846103
x35=53.3696049818501x_{35} = 53.3696049818501
x35=91.0842301384618x_{35} = 91.0842301384618
x35=40.7916847146183x_{35} = 40.7916847146183
x35=−103.653263067797x_{35} = -103.653263067797
x35=65.943118880897x_{35} = 65.943118880897
x35=21.8998872970823x_{35} = 21.8998872970823
x35=−40.7916847146183x_{35} = -40.7916847146183
x35=47.0814165846103x_{35} = 47.0814165846103
x35=−59.656738426191x_{35} = -59.656738426191
x35=−84.7994176724893x_{35} = -84.7994176724893
x35=−2.33112237041442x_{35} = -2.33112237041442
x35=15.5797675022891x_{35} = 15.5797675022891
x35=−72.2289430706097x_{35} = -72.2289430706097
x35=−91.0842301384618x_{35} = -91.0842301384618
x35=−65.943118880897x_{35} = -65.943118880897
x35=−28.2034502671317x_{35} = -28.2034502671317
Decrece en los intervalos
[131.93173238582,∞)\left[131.93173238582, \infty\right)
Crece en los intervalos
(−∞,−100.511067265113]\left(-\infty, -100.511067265113\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=−92.6337941106803x_{1} = -92.6337941106803
x2=23.3904159389741x_{2} = 23.3904159389741
x3=479.084530354542x_{3} = 479.084530354542
x4=45.4650429364822x_{4} = 45.4650429364822
x5=−73.7731905236106x_{5} = -73.7731905236106
x6=−54.9049780507868x_{6} = -54.9049780507868
x7=67.4849478949614x_{7} = 67.4849478949614
x8=−42.3168880500391x_{8} = -42.3168880500391
x9=17.0427193207795x_{9} = 17.0427193207795
x10=−95.7768045829418x_{10} = -95.7768045829418
x11=48.6123445281032x_{11} = 48.6123445281032
x12=−39.1676723486677x_{12} = -39.1676723486677
x13=−17.0427193207795x_{13} = -17.0427193207795
x14=−48.6123445281032x_{14} = -48.6123445281032
x15=83.2041192072701x_{15} = 83.2041192072701
x16=51.758949425305x_{16} = 51.758949425305
x17=42.3168880500391x_{17} = 42.3168880500391
x18=58.0505245424602x_{18} = 58.0505245424602
x19=61.1956635639228x_{19} = 61.1956635639228
x20=−89.4906839379977x_{20} = -89.4906839379977
x21=−13.8457897593825x_{21} = -13.8457897593825
x22=−70.6291819782668x_{22} = -70.6291819782668
x23=39.1676723486677x_{23} = 39.1676723486677
x24=−45.4650429364822x_{24} = -45.4650429364822
x25=−146.05666962229x_{25} = -146.05666962229
x26=−86.3474631570568x_{26} = -86.3474631570568
x27=7.28880961897998x_{27} = 7.28880961897998
x28=−83.2041192072701x_{28} = -83.2041192072701
x29=3.4244262943983x_{29} = 3.4244262943983
x30=−98.9197248732035x_{30} = -98.9197248732035
x31=36.0171147693414x_{31} = 36.0171147693414
x32=73.7731905236106x_{32} = 73.7731905236106
x33=−108.34802323762x_{33} = -108.34802323762
x34=−23.3904159389741x_{34} = -23.3904159389741
x35=−80.0606375496603x_{35} = -80.0606375496603
x36=−7.28880961897998x_{36} = -7.28880961897998
x37=95.7768045829418x_{37} = 95.7768045829418
x38=76.9170012609926x_{38} = 76.9170012609926
x39=54.9049780507868x_{39} = 54.9049780507868
x40=−64.3404550990583x_{40} = -64.3404550990583
x41=86.3474631570568x_{41} = 86.3474631570568
x42=−32.8648247785944x_{42} = -32.8648247785944
x43=20.2217437216905x_{43} = 20.2217437216905
x44=2430.02027147444x_{44} = 2430.02027147444
x45=32.8648247785944x_{45} = 32.8648247785944
x46=10.6132292669714x_{46} = 10.6132292669714
x47=64.3404550990583x_{47} = 64.3404550990583
x48=−51.758949425305x_{48} = -51.758949425305
x49=−26.5525378692192x_{49} = -26.5525378692192
x50=92.6337941106803x_{50} = 92.6337941106803
x51=70.6291819782668x_{51} = 70.6291819782668
x52=196.329166021576x_{52} = 196.329166021576
x53=13.8457897593825x_{53} = 13.8457897593825
x54=98.9197248732035x_{54} = 98.9197248732035
x55=−58.0505245424602x_{55} = -58.0505245424602
x56=−29.710242960192x_{56} = -29.710242960192
x57=29.710242960192x_{57} = 29.710242960192
x58=−67.4849478949614x_{58} = -67.4849478949614
x59=−76.9170012609926x_{59} = -76.9170012609926
x60=−20.2217437216905x_{60} = -20.2217437216905
x61=89.4906839379977x_{61} = 89.4906839379977
x62=−61.1956635639228x_{62} = -61.1956635639228
x63=−10.6132292669714x_{63} = -10.6132292669714
x64=26.5525378692192x_{64} = 26.5525378692192
x65=80.0606375496603x_{65} = 80.0606375496603
x66=−3.4244262943983x_{66} = -3.4244262943983
x67=−36.0171147693414x_{67} = -36.0171147693414
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
x2=1x_{2} = 1

lim⁡x→−1−(−4xsin⁡(x)x2−1+cos⁡(x)−2(4x2x2−1−1)cos⁡(x)x2−1x2−1)=−∞\lim_{x \to -1^-}\left(\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}\right) = -\infty
lim⁡x→−1+(−4xsin⁡(x)x2−1+cos⁡(x)−2(4x2x2−1−1)cos⁡(x)x2−1x2−1)=∞\lim_{x \to -1^+}\left(\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}\right) = \infty
- los límites no son iguales, signo
x1=−1x_{1} = -1
- es el punto de flexión
lim⁡x→1−(−4xsin⁡(x)x2−1+cos⁡(x)−2(4x2x2−1−1)cos⁡(x)x2−1x2−1)=∞\lim_{x \to 1^-}\left(\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}\right) = \infty
lim⁡x→1+(−4xsin⁡(x)x2−1+cos⁡(x)−2(4x2x2−1−1)cos⁡(x)x2−1x2−1)=−∞\lim_{x \to 1^+}\left(\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}\right) = -\infty
- los límites no son iguales, signo
x2=1x_{2} = 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
[2430.02027147444,∞)\left[2430.02027147444, \infty\right)
Convexa en los intervalos
(−∞,−146.05666962229]\left(-\infty, -146.05666962229\right]
Asíntotas verticales
Hay:
x1=−1x_{1} = -1
x2=1x_{2} = 1
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(cos⁡(x)1−x2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{1 - 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)1−x2)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{1 - 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)/(1 - x^2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x)x(1−x2))=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x \left(1 - x^{2}\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x)x(1−x2))=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x \left(1 - x^{2}\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)1−x2=cos⁡(x)1−x2\frac{\cos{\left(x \right)}}{1 - x^{2}} = \frac{\cos{\left(x \right)}}{1 - x^{2}}
- Sí
cos⁡(x)1−x2=−cos⁡(x)1−x2\frac{\cos{\left(x \right)}}{1 - x^{2}} = - \frac{\cos{\left(x \right)}}{1 - x^{2}}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор