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Trazado el gráfico de la función/ xsinx+(e^x-e^x)/(e^-x+e^x)

Gráfico de la función y = xsinx+(e^x-e^x)/(e^-x+e^x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                   x    x 
                  E  - E  
f(x) = x*sin(x) + --------
                   -x    x
                  E   + E
f(x)=xsin⁡(x)+−ex+exex+e−xf{\left(x \right)} = x \sin{\left(x \right)} + \frac{- e^{x} + e^{x}}{e^{x} + e^{- x}} 
f = x*sin(x) + (-E^x + E^x)/(E^x + E^(-x))
Gráfico de la función
02468-8-6-4-2-1010-1010
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:
xsin⁡(x)+−ex+exex+e−x=0x \sin{\left(x \right)} + \frac{- e^{x} + e^{x}}{e^{x} + e^{- x}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=12.5663706143592x_{1} = 12.5663706143592
x2=53.4070751110265x_{2} = 53.4070751110265
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=37.6991118430775x_{4} = 37.6991118430775
x5=97.3893722612836x_{5} = 97.3893722612836
x6=78.5398163397448x_{6} = 78.5398163397448
x7=−59.6902604182061x_{7} = -59.6902604182061
x8=−65.9734457253857x_{8} = -65.9734457253857
x9=0x_{9} = 0
x10=−31.4159265358979x_{10} = -31.4159265358979
x11=−50.2654824574367x_{11} = -50.2654824574367
x12=−21.9911485751286x_{12} = -21.9911485751286
x13=6.28318530717959x_{13} = 6.28318530717959
x14=−34.5575191894877x_{14} = -34.5575191894877
x15=69.1150383789755x_{15} = 69.1150383789755
x16=−94.2477796076938x_{16} = -94.2477796076938
x17=−69.1150383789755x_{17} = -69.1150383789755
x18=−15.707963267949x_{18} = -15.707963267949
x19=21.9911485751286x_{19} = 21.9911485751286
x20=62.8318530717959x_{20} = 62.8318530717959
x21=50.2654824574367x_{21} = 50.2654824574367
x22=81.6814089933346x_{22} = 81.6814089933346
x23=100.530964914873x_{23} = 100.530964914873
x24=−40.8407044966673x_{24} = -40.8407044966673
x25=9.42477796076938x_{25} = 9.42477796076938
x26=−87.9645943005142x_{26} = -87.9645943005142
x27=34.5575191894877x_{27} = 34.5575191894877
x28=65.9734457253857x_{28} = 65.9734457253857
x29=−62.8318530717959x_{29} = -62.8318530717959
x30=−18.8495559215388x_{30} = -18.8495559215388
x31=−28.2743338823081x_{31} = -28.2743338823081
x32=−56.5486677646163x_{32} = -56.5486677646163
x33=−53.4070751110265x_{33} = -53.4070751110265
x34=−37.6991118430775x_{34} = -37.6991118430775
x35=−25.1327412287183x_{35} = -25.1327412287183
x36=−100.530964914873x_{36} = -100.530964914873
x37=−9.42477796076938x_{37} = -9.42477796076938
x38=40.8407044966673x_{38} = 40.8407044966673
x39=−91.106186954104x_{39} = -91.106186954104
x40=−75.398223686155x_{40} = -75.398223686155
x41=18.8495559215388x_{41} = 18.8495559215388
x42=87.9645943005142x_{42} = 87.9645943005142
x43=59.6902604182061x_{43} = 59.6902604182061
x44=−6.28318530717959x_{44} = -6.28318530717959
x45=25.1327412287183x_{45} = 25.1327412287183
x46=47.1238898038469x_{46} = 47.1238898038469
x47=697.433569096934x_{47} = 697.433569096934
x48=91.106186954104x_{48} = 91.106186954104
x49=28.2743338823081x_{49} = 28.2743338823081
x50=56.5486677646163x_{50} = 56.5486677646163
x51=−43.9822971502571x_{51} = -43.9822971502571
x52=−47.1238898038469x_{52} = -47.1238898038469
x53=−3.14159265358979x_{53} = -3.14159265358979
x54=31.4159265358979x_{54} = 31.4159265358979
x55=94.2477796076938x_{55} = 94.2477796076938
x56=−12.5663706143592x_{56} = -12.5663706143592
x57=75.398223686155x_{57} = 75.398223686155
x58=−72.2566310325652x_{58} = -72.2566310325652
x59=−84.8230016469244x_{59} = -84.8230016469244
x60=84.8230016469244x_{60} = 84.8230016469244
x61=72.2566310325652x_{61} = 72.2566310325652
x62=−81.6814089933346x_{62} = -81.6814089933346
x63=43.9822971502571x_{63} = 43.9822971502571
x64=−78.5398163397448x_{64} = -78.5398163397448
x65=15.707963267949x_{65} = 15.707963267949
x66=3.14159265358979x_{66} = 3.14159265358979
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 x*sin(x) + (E^x - E^x)/(E^(-x) + E^x).
0sin⁡(0)+−e0+e0e−0+e00 \sin{\left(0 \right)} + \frac{- e^{0} + e^{0}}{e^{- 0} + e^{0}}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
xcos⁡(x)+(−ex+ex)(−ex+e−x)(ex+e−x)2+sin⁡(x)+ex−exex+e−x=0x \cos{\left(x \right)} + \frac{\left(- e^{x} + e^{x}\right) \left(- e^{x} + e^{- x}\right)}{\left(e^{x} + e^{- x}\right)^{2}} + \sin{\left(x \right)} + \frac{e^{x} - e^{x}}{e^{x} + e^{- x}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=48.7152107175577x_{1} = 48.7152107175577
x2=33.0170010333572x_{2} = 33.0170010333572
x3=76.9820093304187x_{3} = 76.9820093304187
x4=−39.295350981473x_{4} = -39.295350981473
x5=45.57503179559x_{5} = 45.57503179559
x6=42.4350618814099x_{6} = 42.4350618814099
x7=51.855560729152x_{7} = 51.855560729152
x8=−36.1559664195367x_{8} = -36.1559664195367
x9=−42.4350618814099x_{9} = -42.4350618814099
x10=−2.02875783811043x_{10} = -2.02875783811043
x11=−26.7409160147873x_{11} = -26.7409160147873
x12=0x_{12} = 0
x13=4.91318043943488x_{13} = 4.91318043943488
x14=83.2642147040886x_{14} = 83.2642147040886
x15=−23.6042847729804x_{15} = -23.6042847729804
x16=−92.687771772017x_{16} = -92.687771772017
x17=64.4181717218392x_{17} = 64.4181717218392
x18=73.8409691490209x_{18} = 73.8409691490209
x19=−95.8290108090195x_{19} = -95.8290108090195
x20=−83.2642147040886x_{20} = -83.2642147040886
x21=29.8785865061074x_{21} = 29.8785865061074
x22=−89.5465575382492x_{22} = -89.5465575382492
x23=17.3363779239834x_{23} = 17.3363779239834
x24=−86.4053708116885x_{24} = -86.4053708116885
x25=11.085538406497x_{25} = 11.085538406497
x26=−4.91318043943488x_{26} = -4.91318043943488
x27=7.97866571241324x_{27} = 7.97866571241324
x28=−29.8785865061074x_{28} = -29.8785865061074
x29=−11.085538406497x_{29} = -11.085538406497
x30=−98.9702722883957x_{30} = -98.9702722883957
x31=−17.3363779239834x_{31} = -17.3363779239834
x32=98.9702722883957x_{32} = 98.9702722883957
x33=54.9960525574964x_{33} = 54.9960525574964
x34=102.111554139654x_{34} = 102.111554139654
x35=20.469167402741x_{35} = 20.469167402741
x36=58.1366632448992x_{36} = 58.1366632448992
x37=39.295350981473x_{37} = 39.295350981473
x38=−64.4181717218392x_{38} = -64.4181717218392
x39=67.5590428388084x_{39} = 67.5590428388084
x40=−51.855560729152x_{40} = -51.855560729152
x41=86.4053708116885x_{41} = 86.4053708116885
x42=61.2773745335697x_{42} = 61.2773745335697
x43=−80.1230928148503x_{43} = -80.1230928148503
x44=80.1230928148503x_{44} = 80.1230928148503
x45=−76.9820093304187x_{45} = -76.9820093304187
x46=−73.8409691490209x_{46} = -73.8409691490209
x47=−67.5590428388084x_{47} = -67.5590428388084
x48=−70.69997803861x_{48} = -70.69997803861
x49=95.8290108090195x_{49} = 95.8290108090195
x50=92.687771772017x_{50} = 92.687771772017
x51=−48.7152107175577x_{51} = -48.7152107175577
x52=−61.2773745335697x_{52} = -61.2773745335697
x53=2.02875783811043x_{53} = 2.02875783811043
x54=26.7409160147873x_{54} = 26.7409160147873
x55=−33.0170010333572x_{55} = -33.0170010333572
x56=14.2074367251912x_{56} = 14.2074367251912
x57=−58.1366632448992x_{57} = -58.1366632448992
x58=70.69997803861x_{58} = 70.69997803861
x59=89.5465575382492x_{59} = 89.5465575382492
x60=−7.97866571241324x_{60} = -7.97866571241324
x61=−54.9960525574964x_{61} = -54.9960525574964
x62=36.1559664195367x_{62} = 36.1559664195367
x63=−20.469167402741x_{63} = -20.469167402741
x64=−45.57503179559x_{64} = -45.57503179559
x65=−14.2074367251912x_{65} = -14.2074367251912
x66=23.6042847729804x_{66} = 23.6042847729804
Signos de extremos en los puntos:
(48.715210717557724, -48.7049502253679)

(33.017001033357246, 33.0018677308454)

(76.98200933041872, 76.9755151282637)

(-39.295350981472986, 39.2826330068918)

(45.57503179559002, 45.5640648360268)

(42.43506188140989, -42.4232840772591)

(51.85556072915197, 51.8459212502015)

(-36.15596641953672, -36.1421453722421)

(-42.43506188140989, -42.4232840772591)

(-2.028757838110434, 1.81970574115965)

(-26.74091601478731, 26.7222376646974)

(0, 0)

(4.913180439434884, -4.81446988971227)

(83.26421470408864, 83.2582103729533)

(-23.604284772980407, -23.5831306496334)

(-92.687771772017, -92.6823777880592)

(64.41817172183916, 64.4104113393753)

(73.8409691490209, -73.8341987715416)

(-95.82901080901948, 95.8237936084657)

(-83.26421470408864, 83.2582103729533)

(29.878586506107393, -29.8618661591868)

(-89.54655753824919, 89.5409743728852)

(17.33637792398336, -17.3076086078585)

(-86.40537081168854, -86.3995847156108)

(11.085538406497022, -11.04070801593)

(-4.913180439434884, -4.81446988971227)

(7.978665712413241, 7.91672737158778)

(-29.878586506107393, -29.8618661591868)

(-11.085538406497022, -11.04070801593)

(-98.9702722883957, -98.9652206531187)

(-17.33637792398336, -17.3076086078585)

(98.9702722883957, -98.9652206531187)

(54.99605255749639, -54.9869632496976)

(102.11155413965392, 102.106657886316)

(20.46916740274095, 20.4447840582523)

(58.13666324489916, 58.1280647280857)

(39.295350981472986, 39.2826330068918)

(-64.41817172183916, 64.4104113393753)

(67.5590428388084, -67.5516431209725)

(-51.85556072915197, 51.8459212502015)

(86.40537081168854, -86.3995847156108)

(61.277374533569656, -61.2692165444766)

(-80.12309281485025, -80.1168531456592)

(80.12309281485025, -80.1168531456592)

(-76.98200933041872, 76.9755151282637)

(-73.8409691490209, -73.8341987715416)

(-67.5590428388084, -67.5516431209725)

(-70.69997803861, 70.6929069615931)

(95.82901080901948, 95.8237936084657)

(92.687771772017, -92.6823777880592)

(-48.715210717557724, -48.7049502253679)

(-61.277374533569656, -61.2692165444766)

(2.028757838110434, 1.81970574115965)

(26.74091601478731, 26.7222376646974)

(-33.017001033357246, 33.0018677308454)

(14.207436725191188, 14.1723741137743)

(-58.13666324489916, 58.1280647280857)

(70.69997803861, 70.6929069615931)

(89.54655753824919, 89.5409743728852)

(-7.978665712413241, 7.91672737158778)

(-54.99605255749639, -54.9869632496976)

(36.15596641953672, -36.1421453722421)

(-20.46916740274095, 20.4447840582523)

(-45.57503179559002, 45.5640648360268)

(-14.207436725191188, 14.1723741137743)

(23.604284772980407, -23.5831306496334)


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=48.7152107175577x_{1} = 48.7152107175577
x2=42.4350618814099x_{2} = 42.4350618814099
x3=−36.1559664195367x_{3} = -36.1559664195367
x4=−42.4350618814099x_{4} = -42.4350618814099
x5=0x_{5} = 0
x6=4.91318043943488x_{6} = 4.91318043943488
x7=−23.6042847729804x_{7} = -23.6042847729804
x8=−92.687771772017x_{8} = -92.687771772017
x9=73.8409691490209x_{9} = 73.8409691490209
x10=29.8785865061074x_{10} = 29.8785865061074
x11=17.3363779239834x_{11} = 17.3363779239834
x12=−86.4053708116885x_{12} = -86.4053708116885
x13=11.085538406497x_{13} = 11.085538406497
x14=−4.91318043943488x_{14} = -4.91318043943488
x15=−29.8785865061074x_{15} = -29.8785865061074
x16=−11.085538406497x_{16} = -11.085538406497
x17=−98.9702722883957x_{17} = -98.9702722883957
x18=−17.3363779239834x_{18} = -17.3363779239834
x19=98.9702722883957x_{19} = 98.9702722883957
x20=54.9960525574964x_{20} = 54.9960525574964
x21=67.5590428388084x_{21} = 67.5590428388084
x22=86.4053708116885x_{22} = 86.4053708116885
x23=61.2773745335697x_{23} = 61.2773745335697
x24=−80.1230928148503x_{24} = -80.1230928148503
x25=80.1230928148503x_{25} = 80.1230928148503
x26=−73.8409691490209x_{26} = -73.8409691490209
x27=−67.5590428388084x_{27} = -67.5590428388084
x28=92.687771772017x_{28} = 92.687771772017
x29=−48.7152107175577x_{29} = -48.7152107175577
x30=−61.2773745335697x_{30} = -61.2773745335697
x31=−54.9960525574964x_{31} = -54.9960525574964
x32=36.1559664195367x_{32} = 36.1559664195367
x33=23.6042847729804x_{33} = 23.6042847729804
Puntos máximos de la función:
x33=33.0170010333572x_{33} = 33.0170010333572
x33=76.9820093304187x_{33} = 76.9820093304187
x33=−39.295350981473x_{33} = -39.295350981473
x33=45.57503179559x_{33} = 45.57503179559
x33=51.855560729152x_{33} = 51.855560729152
x33=−2.02875783811043x_{33} = -2.02875783811043
x33=−26.7409160147873x_{33} = -26.7409160147873
x33=83.2642147040886x_{33} = 83.2642147040886
x33=64.4181717218392x_{33} = 64.4181717218392
x33=−95.8290108090195x_{33} = -95.8290108090195
x33=−83.2642147040886x_{33} = -83.2642147040886
x33=−89.5465575382492x_{33} = -89.5465575382492
x33=7.97866571241324x_{33} = 7.97866571241324
x33=102.111554139654x_{33} = 102.111554139654
x33=20.469167402741x_{33} = 20.469167402741
x33=58.1366632448992x_{33} = 58.1366632448992
x33=39.295350981473x_{33} = 39.295350981473
x33=−64.4181717218392x_{33} = -64.4181717218392
x33=−51.855560729152x_{33} = -51.855560729152
x33=−76.9820093304187x_{33} = -76.9820093304187
x33=−70.69997803861x_{33} = -70.69997803861
x33=95.8290108090195x_{33} = 95.8290108090195
x33=2.02875783811043x_{33} = 2.02875783811043
x33=26.7409160147873x_{33} = 26.7409160147873
x33=−33.0170010333572x_{33} = -33.0170010333572
x33=14.2074367251912x_{33} = 14.2074367251912
x33=−58.1366632448992x_{33} = -58.1366632448992
x33=70.69997803861x_{33} = 70.69997803861
x33=89.5465575382492x_{33} = 89.5465575382492
x33=−7.97866571241324x_{33} = -7.97866571241324
x33=−20.469167402741x_{33} = -20.469167402741
x33=−45.57503179559x_{33} = -45.57503179559
x33=−14.2074367251912x_{33} = -14.2074367251912
Decrece en los intervalos
[98.9702722883957,∞)\left[98.9702722883957, \infty\right)
Crece en los intervalos
(−∞,−98.9702722883957]\left(-\infty, -98.9702722883957\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
−xsin⁡(x)+2cos⁡(x)=0- x \sin{\left(x \right)} + 2 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−40.8895777660408x_{1} = -40.8895777660408
x2=−84.8465692433091x_{2} = -84.8465692433091
x3=72.2842925036825x_{3} = 72.2842925036825
x4=−50.3052188363296x_{4} = -50.3052188363296
x5=15.8336114149477x_{5} = 15.8336114149477
x6=−6.57833373272234x_{6} = -6.57833373272234
x7=−25.2119030642106x_{7} = -25.2119030642106
x8=40.8895777660408x_{8} = 40.8895777660408
x9=69.1439554764926x_{9} = 69.1439554764926
x10=−91.1281305511393x_{10} = -91.1281305511393
x11=−9.62956034329743x_{11} = -9.62956034329743
x12=−56.5839987378634x_{12} = -56.5839987378634
x13=25.2119030642106x_{13} = 25.2119030642106
x14=−81.7058821480364x_{14} = -81.7058821480364
x15=53.4444796697636x_{15} = 53.4444796697636
x16=66.0037377708277x_{16} = 66.0037377708277
x17=−75.4247339745236x_{17} = -75.4247339745236
x18=31.479374920314x_{18} = 31.479374920314
x19=75.4247339745236x_{19} = 75.4247339745236
x20=91.1281305511393x_{20} = 91.1281305511393
x21=−37.7520396346102x_{21} = -37.7520396346102
x22=−12.7222987717666x_{22} = -12.7222987717666
x23=−3.6435971674254x_{23} = -3.6435971674254
x24=−18.954681766529x_{24} = -18.954681766529
x25=50.3052188363296x_{25} = 50.3052188363296
x26=−69.1439554764926x_{26} = -69.1439554764926
x27=34.6152330552306x_{27} = 34.6152330552306
x28=−94.2689923093066x_{28} = -94.2689923093066
x29=22.0814757672807x_{29} = 22.0814757672807
x30=−62.863657228703x_{30} = -62.863657228703
x31=37.7520396346102x_{31} = 37.7520396346102
x32=56.5839987378634x_{32} = 56.5839987378634
x33=−34.6152330552306x_{33} = -34.6152330552306
x34=−97.4099011706723x_{34} = -97.4099011706723
x35=94.2689923093066x_{35} = 94.2689923093066
x36=59.7237354324305x_{36} = 59.7237354324305
x37=−44.0276918992479x_{37} = -44.0276918992479
x38=28.3447768697864x_{38} = 28.3447768697864
x39=100.550852725424x_{39} = 100.550852725424
x40=1.0768739863118x_{40} = 1.0768739863118
x41=6.57833373272234x_{41} = 6.57833373272234
x42=−59.7237354324305x_{42} = -59.7237354324305
x43=84.8465692433091x_{43} = 84.8465692433091
x44=−1.0768739863118x_{44} = -1.0768739863118
x45=−47.1662676027767x_{45} = -47.1662676027767
x46=47.1662676027767x_{46} = 47.1662676027767
x47=−78.5652673845995x_{47} = -78.5652673845995
x48=−66.0037377708277x_{48} = -66.0037377708277
x49=3.6435971674254x_{49} = 3.6435971674254
x50=9.62956034329743x_{50} = 9.62956034329743
x51=−128.820822990274x_{51} = -128.820822990274
x52=−100.550852725424x_{52} = -100.550852725424
x53=−28.3447768697864x_{53} = -28.3447768697864
x54=−15.8336114149477x_{54} = -15.8336114149477
x55=78.5652673845995x_{55} = 78.5652673845995
x56=87.9873209346887x_{56} = 87.9873209346887
x57=81.7058821480364x_{57} = 81.7058821480364
x58=97.4099011706723x_{58} = 97.4099011706723
x59=−31.479374920314x_{59} = -31.479374920314
x60=−87.9873209346887x_{60} = -87.9873209346887
x61=−22.0814757672807x_{61} = -22.0814757672807
x62=−72.2842925036825x_{62} = -72.2842925036825
x63=−53.4444796697636x_{63} = -53.4444796697636
x64=44.0276918992479x_{64} = 44.0276918992479
x65=18.954681766529x_{65} = 18.954681766529
x66=62.863657228703x_{66} = 62.863657228703
x67=12.7222987717666x_{67} = 12.7222987717666

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
[97.4099011706723,∞)\left[97.4099011706723, \infty\right)
Convexa en los intervalos
(−∞,−100.550852725424]\left(-\infty, -100.550852725424\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xsin⁡(x)+−ex+exex+e−x)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \sin{\left(x \right)} + \frac{- e^{x} + e^{x}}{e^{x} + e^{- x}}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(xsin⁡(x)+−ex+exex+e−x)=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \sin{\left(x \right)} + \frac{- e^{x} + e^{x}}{e^{x} + e^{- x}}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x*sin(x) + (E^x - E^x)/(E^(-x) + E^x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(xsin⁡(x)+−ex+exex+e−xx)=⟨−1,1⟩\lim_{x \to -\infty}\left(\frac{x \sin{\left(x \right)} + \frac{- e^{x} + e^{x}}{e^{x} + e^{- x}}}{x}\right) = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
lim⁡x→∞(xsin⁡(x)+−ex+exex+e−xx)=⟨−1,1⟩\lim_{x \to \infty}\left(\frac{x \sin{\left(x \right)} + \frac{- e^{x} + e^{x}}{e^{x} + e^{- x}}}{x}\right) = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
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:
xsin⁡(x)+−ex+exex+e−x=xsin⁡(x)x \sin{\left(x \right)} + \frac{- e^{x} + e^{x}}{e^{x} + e^{- x}} = x \sin{\left(x \right)}
- No
xsin⁡(x)+−ex+exex+e−x=−xsin⁡(x)x \sin{\left(x \right)} + \frac{- e^{x} + e^{x}}{e^{x} + e^{- x}} = - x \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор