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Trazado el gráfico de la función/ sin(x)/((3*x))

Gráfico de la función y = sin(x)/((3*x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       sin(x)
f(x) = ------
        3*x
f(x)=sin⁡(x)3xf{\left(x \right)} = \frac{\sin{\left(x \right)}}{3 x} 
f = sin(x)/((3*x))
Gráfico de la función
02468-8-6-4-2-10100.5-0.5
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
sin⁡(x)3x=0\frac{\sin{\left(x \right)}}{3 x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=πx_{1} = \pi
Solución numérica
x1=−59.6902604182061x_{1} = -59.6902604182061
x2=−62.8318530717959x_{2} = -62.8318530717959
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=87.9645943005142x_{4} = 87.9645943005142
x5=−56.5486677646163x_{5} = -56.5486677646163
x6=31.4159265358979x_{6} = 31.4159265358979
x7=69.1150383789755x_{7} = 69.1150383789755
x8=−370.707933123596x_{8} = -370.707933123596
x9=−37.6991118430775x_{9} = -37.6991118430775
x10=−81.6814089933346x_{10} = -81.6814089933346
x11=153.9380400259x_{11} = 153.9380400259
x12=−84.8230016469244x_{12} = -84.8230016469244
x13=−21.9911485751286x_{13} = -21.9911485751286
x14=47.1238898038469x_{14} = 47.1238898038469
x15=537.212343763855x_{15} = 537.212343763855
x16=−223.053078404875x_{16} = -223.053078404875
x17=−113.097335529233x_{17} = -113.097335529233
x18=−12.5663706143592x_{18} = -12.5663706143592
x19=12.5663706143592x_{19} = 12.5663706143592
x20=−87.9645943005142x_{20} = -87.9645943005142
x21=53.4070751110265x_{21} = 53.4070751110265
x22=−15.707963267949x_{22} = -15.707963267949
x23=−100.530964914873x_{23} = -100.530964914873
x24=−3.14159265358979x_{24} = -3.14159265358979
x25=72.2566310325652x_{25} = 72.2566310325652
x26=34.5575191894877x_{26} = 34.5575191894877
x27=−94.2477796076938x_{27} = -94.2477796076938
x28=6.28318530717959x_{28} = 6.28318530717959
x29=−69.1150383789755x_{29} = -69.1150383789755
x30=97.3893722612836x_{30} = 97.3893722612836
x31=65.9734457253857x_{31} = 65.9734457253857
x32=590.619418874881x_{32} = 590.619418874881
x33=−50.2654824574367x_{33} = -50.2654824574367
x34=15.707963267949x_{34} = 15.707963267949
x35=3.14159265358979x_{35} = 3.14159265358979
x36=−25.1327412287183x_{36} = -25.1327412287183
x37=−18.8495559215388x_{37} = -18.8495559215388
x38=40.8407044966673x_{38} = 40.8407044966673
x39=−53.4070751110265x_{39} = -53.4070751110265
x40=37.6991118430775x_{40} = 37.6991118430775
x41=−43.9822971502571x_{41} = -43.9822971502571
x42=18.8495559215388x_{42} = 18.8495559215388
x43=−78.5398163397448x_{43} = -78.5398163397448
x44=−6.28318530717959x_{44} = -6.28318530717959
x45=−40.8407044966673x_{45} = -40.8407044966673
x46=43.9822971502571x_{46} = 43.9822971502571
x47=56.5486677646163x_{47} = 56.5486677646163
x48=−65.9734457253857x_{48} = -65.9734457253857
x49=3647.38907081741x_{49} = 3647.38907081741
x50=25.1327412287183x_{50} = 25.1327412287183
x51=78.5398163397448x_{51} = 78.5398163397448
x52=−28.2743338823081x_{52} = -28.2743338823081
x53=75.398223686155x_{53} = 75.398223686155
x54=59.6902604182061x_{54} = 59.6902604182061
x55=−34.5575191894877x_{55} = -34.5575191894877
x56=81.6814089933346x_{56} = 81.6814089933346
x57=−47.1238898038469x_{57} = -47.1238898038469
x58=100.530964914873x_{58} = 100.530964914873
x59=−9.42477796076938x_{59} = -9.42477796076938
x60=−75.398223686155x_{60} = -75.398223686155
x61=−72.2566310325652x_{61} = -72.2566310325652
x62=−31.4159265358979x_{62} = -31.4159265358979
x63=28.2743338823081x_{63} = 28.2743338823081
x64=−91.106186954104x_{64} = -91.106186954104
x65=21.9911485751286x_{65} = 21.9911485751286
x66=62.8318530717959x_{66} = 62.8318530717959
x67=9.42477796076938x_{67} = 9.42477796076938
x68=50.2654824574367x_{68} = 50.2654824574367
x69=94.2477796076938x_{69} = 94.2477796076938
x70=91.106186954104x_{70} = 91.106186954104
x71=84.8230016469244x_{71} = 84.8230016469244
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 sin(x)/((3*x)).
sin⁡(0)0⋅3\frac{\sin{\left(0 \right)}}{0 \cdot 3}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
13xcos⁡(x)−sin⁡(x)3x2=0\frac{1}{3 x} \cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{3 x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=17.2207552719308x_{1} = 17.2207552719308
x2=−80.0981286289451x_{2} = -80.0981286289451
x3=−83.2401924707234x_{3} = -83.2401924707234
x4=98.9500628243319x_{4} = 98.9500628243319
x5=−45.5311340139913x_{5} = -45.5311340139913
x6=−86.3822220347287x_{6} = -86.3822220347287
x7=7.72525183693771x_{7} = 7.72525183693771
x8=−394.267341680887x_{8} = -394.267341680887
x9=−4.49340945790906x_{9} = -4.49340945790906
x10=4.49340945790906x_{10} = 4.49340945790906
x11=108.375719651675x_{11} = 108.375719651675
x12=39.2444323611642x_{12} = 39.2444323611642
x13=−70.6716857116195x_{13} = -70.6716857116195
x14=10.9041216594289x_{14} = 10.9041216594289
x15=−42.3879135681319x_{15} = -42.3879135681319
x16=80.0981286289451x_{16} = 80.0981286289451
x17=89.5242209304172x_{17} = 89.5242209304172
x18=−48.6741442319544x_{18} = -48.6741442319544
x19=14.0661939128315x_{19} = 14.0661939128315
x20=−36.1006222443756x_{20} = -36.1006222443756
x21=−95.8081387868617x_{21} = -95.8081387868617
x22=64.3871195905574x_{22} = 64.3871195905574
x23=61.2447302603744x_{23} = 61.2447302603744
x24=−54.9596782878889x_{24} = -54.9596782878889
x25=76.9560263103312x_{25} = 76.9560263103312
x26=−76.9560263103312x_{26} = -76.9560263103312
x27=−98.9500628243319x_{27} = -98.9500628243319
x28=−7.72525183693771x_{28} = -7.72525183693771
x29=−20.3713029592876x_{29} = -20.3713029592876
x30=−39.2444323611642x_{30} = -39.2444323611642
x31=−14.0661939128315x_{31} = -14.0661939128315
x32=−32.9563890398225x_{32} = -32.9563890398225
x33=54.9596782878889x_{33} = 54.9596782878889
x34=73.8138806006806x_{34} = 73.8138806006806
x35=26.6660542588127x_{35} = 26.6660542588127
x36=4120.19852247627x_{36} = 4120.19852247627
x37=−26.6660542588127x_{37} = -26.6660542588127
x38=−61.2447302603744x_{38} = -61.2447302603744
x39=−67.5294347771441x_{39} = -67.5294347771441
x40=29.811598790893x_{40} = 29.811598790893
x41=51.8169824872797x_{41} = 51.8169824872797
x42=23.519452498689x_{42} = 23.519452498689
x43=−58.1022547544956x_{43} = -58.1022547544956
x44=67.5294347771441x_{44} = 67.5294347771441
x45=−10.9041216594289x_{45} = -10.9041216594289
x46=−89.5242209304172x_{46} = -89.5242209304172
x47=86.3822220347287x_{47} = 86.3822220347287
x48=−23.519452498689x_{48} = -23.519452498689
x49=−17.2207552719308x_{49} = -17.2207552719308
x50=58.1022547544956x_{50} = 58.1022547544956
x51=2436.30469240122x_{51} = 2436.30469240122
x52=−92.6661922776228x_{52} = -92.6661922776228
x53=−29.811598790893x_{53} = -29.811598790893
x54=92.6661922776228x_{54} = 92.6661922776228
x55=−64.3871195905574x_{55} = -64.3871195905574
x56=32.9563890398225x_{56} = 32.9563890398225
x57=20.3713029592876x_{57} = 20.3713029592876
x58=48.6741442319544x_{58} = 48.6741442319544
x59=45.5311340139913x_{59} = 45.5311340139913
x60=36.1006222443756x_{60} = 36.1006222443756
x61=70.6716857116195x_{61} = 70.6716857116195
x62=83.2401924707234x_{62} = 83.2401924707234
x63=95.8081387868617x_{63} = 95.8081387868617
x64=−73.8138806006806x_{64} = -73.8138806006806
x65=42.3879135681319x_{65} = 42.3879135681319
x66=−51.8169824872797x_{66} = -51.8169824872797
Signos de extremos en los puntos:
(17.22075527193077, -0.0193239341153846)

(-80.09812862894512, -0.00416123777392633)

(-83.2401924707234, 0.00400418682735091)

(98.95006282433188, -0.00336853057883468)

(-45.53113401399128, 0.00731923274282748)

(-86.38222203472871, -0.00385856015282259)

(7.725251836937707, 0.0427915178419664)

(-394.26734168088706, -0.000845447304204278)

(-4.493409457909064, -0.0724112094037406)

(4.493409457909064, -0.0724112094037406)

(108.37571965167469, 0.00307558875026066)

(39.24443236116419, 0.00849101769762692)

(-70.6716857116195, 0.00471617402162213)

(10.904121659428899, -0.0304417342743526)

(-42.38791356813192, -0.00786168940967213)

(80.09812862894512, -0.00416123777392633)

(89.52422093041719, 0.00372315487805785)

(-48.674144231954386, -0.00684681801391791)

(14.066193912831473, 0.0236378198168207)

(-36.10062224437561, -0.00922991076704973)

(-95.8081387868617, 0.00347898604485527)

(64.38711959055742, 0.00517639460248711)

(61.2447302603744, -0.00544191977366594)

(-54.959678287888934, -0.00606404877393438)

(76.95602631033118, 0.00433111232901424)

(-76.95602631033118, 0.00433111232901424)

(-98.95006282433188, -0.00336853057883468)

(-7.725251836937707, 0.0427915178419664)

(-20.37130295928756, 0.0163432080046914)

(-39.24443236116419, 0.00849101769762692)

(-14.066193912831473, 0.0236378198168207)

(-32.956389039822476, 0.0101097237287701)

(54.959678287888934, -0.00606404877393438)

(73.81388060068065, -0.00451544811504665)

(26.666054258812675, 0.0124915066646437)

(4120.198522476267, -8.09022482041076e-5)

(-26.666054258812675, 0.0124915066646437)

(-61.2447302603744, -0.00544191977366594)

(-67.52943477714412, -0.00493557798218307)

(29.81159879089296, -0.0111750450071329)

(51.81698248727967, 0.00643169982919599)

(23.519452498689006, -0.0141598723258709)

(-58.10225475449559, 0.00573616249054265)

(67.52943477714412, -0.00493557798218307)

(-10.904121659428899, -0.0304417342743526)

(-89.52422093041719, 0.00372315487805785)

(86.38222203472871, -0.00385856015282259)

(-23.519452498689006, -0.0141598723258709)

(-17.22075527193077, -0.0193239341153846)

(58.10225475449559, 0.00573616249054265)

(2436.304692401216, -0.00013681921899742)

(-92.66619227762284, -0.00359693128317808)

(-29.81159879089296, -0.0111750450071329)

(92.66619227762284, -0.00359693128317808)

(-64.38711959055742, 0.00517639460248711)

(32.956389039822476, 0.0101097237287701)

(20.37130295928756, 0.0163432080046914)

(48.674144231954386, -0.00684681801391791)

(45.53113401399128, 0.00731923274282748)

(36.10062224437561, -0.00922991076704973)

(70.6716857116195, 0.00471617402162213)

(83.2401924707234, 0.00400418682735091)

(95.8081387868617, 0.00347898604485527)

(-73.81388060068065, -0.00451544811504665)

(42.38791356813192, -0.00786168940967213)

(-51.81698248727967, 0.00643169982919599)


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=17.2207552719308x_{1} = 17.2207552719308
x2=−80.0981286289451x_{2} = -80.0981286289451
x3=98.9500628243319x_{3} = 98.9500628243319
x4=−86.3822220347287x_{4} = -86.3822220347287
x5=−394.267341680887x_{5} = -394.267341680887
x6=−4.49340945790906x_{6} = -4.49340945790906
x7=4.49340945790906x_{7} = 4.49340945790906
x8=10.9041216594289x_{8} = 10.9041216594289
x9=−42.3879135681319x_{9} = -42.3879135681319
x10=80.0981286289451x_{10} = 80.0981286289451
x11=−48.6741442319544x_{11} = -48.6741442319544
x12=−36.1006222443756x_{12} = -36.1006222443756
x13=61.2447302603744x_{13} = 61.2447302603744
x14=−54.9596782878889x_{14} = -54.9596782878889
x15=−98.9500628243319x_{15} = -98.9500628243319
x16=54.9596782878889x_{16} = 54.9596782878889
x17=73.8138806006806x_{17} = 73.8138806006806
x18=4120.19852247627x_{18} = 4120.19852247627
x19=−61.2447302603744x_{19} = -61.2447302603744
x20=−67.5294347771441x_{20} = -67.5294347771441
x21=29.811598790893x_{21} = 29.811598790893
x22=23.519452498689x_{22} = 23.519452498689
x23=67.5294347771441x_{23} = 67.5294347771441
x24=−10.9041216594289x_{24} = -10.9041216594289
x25=86.3822220347287x_{25} = 86.3822220347287
x26=−23.519452498689x_{26} = -23.519452498689
x27=−17.2207552719308x_{27} = -17.2207552719308
x28=2436.30469240122x_{28} = 2436.30469240122
x29=−92.6661922776228x_{29} = -92.6661922776228
x30=−29.811598790893x_{30} = -29.811598790893
x31=92.6661922776228x_{31} = 92.6661922776228
x32=48.6741442319544x_{32} = 48.6741442319544
x33=36.1006222443756x_{33} = 36.1006222443756
x34=−73.8138806006806x_{34} = -73.8138806006806
x35=42.3879135681319x_{35} = 42.3879135681319
Puntos máximos de la función:
x35=−83.2401924707234x_{35} = -83.2401924707234
x35=−45.5311340139913x_{35} = -45.5311340139913
x35=7.72525183693771x_{35} = 7.72525183693771
x35=108.375719651675x_{35} = 108.375719651675
x35=39.2444323611642x_{35} = 39.2444323611642
x35=−70.6716857116195x_{35} = -70.6716857116195
x35=89.5242209304172x_{35} = 89.5242209304172
x35=14.0661939128315x_{35} = 14.0661939128315
x35=−95.8081387868617x_{35} = -95.8081387868617
x35=64.3871195905574x_{35} = 64.3871195905574
x35=76.9560263103312x_{35} = 76.9560263103312
x35=−76.9560263103312x_{35} = -76.9560263103312
x35=−7.72525183693771x_{35} = -7.72525183693771
x35=−20.3713029592876x_{35} = -20.3713029592876
x35=−39.2444323611642x_{35} = -39.2444323611642
x35=−14.0661939128315x_{35} = -14.0661939128315
x35=−32.9563890398225x_{35} = -32.9563890398225
x35=26.6660542588127x_{35} = 26.6660542588127
x35=−26.6660542588127x_{35} = -26.6660542588127
x35=51.8169824872797x_{35} = 51.8169824872797
x35=−58.1022547544956x_{35} = -58.1022547544956
x35=−89.5242209304172x_{35} = -89.5242209304172
x35=58.1022547544956x_{35} = 58.1022547544956
x35=−64.3871195905574x_{35} = -64.3871195905574
x35=32.9563890398225x_{35} = 32.9563890398225
x35=20.3713029592876x_{35} = 20.3713029592876
x35=45.5311340139913x_{35} = 45.5311340139913
x35=70.6716857116195x_{35} = 70.6716857116195
x35=83.2401924707234x_{35} = 83.2401924707234
x35=95.8081387868617x_{35} = 95.8081387868617
x35=−51.8169824872797x_{35} = -51.8169824872797
Decrece en los intervalos
[4120.19852247627,∞)\left[4120.19852247627, \infty\right)
Crece en los intervalos
(−∞,−394.267341680887]\left(-\infty, -394.267341680887\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
−sin⁡(x)−2cos⁡(x)x+2sin⁡(x)x23x=0\frac{- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{2 \sin{\left(x \right)}}{x^{2}}}{3 x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−12.404445021902x_{1} = -12.404445021902
x2=−47.0813974121542x_{2} = -47.0813974121542
x3=−18.7426455847748x_{3} = -18.7426455847748
x4=−91.0842274914688x_{4} = -91.0842274914688
x5=9.20584014293667x_{5} = 9.20584014293667
x6=−84.7994143922025x_{6} = -84.7994143922025
x7=21.8996964794928x_{7} = 21.8996964794928
x8=−342.42775856009x_{8} = -342.42775856009
x9=−87.9418500396598x_{9} = -87.9418500396598
x10=−1288.05143523817x_{10} = -1288.05143523817
x11=−59.6567290035279x_{11} = -59.6567290035279
x12=100.511065295271x_{12} = 100.511065295271
x13=−43.9367614714198x_{13} = -43.9367614714198
x14=91.0842274914688x_{14} = 91.0842274914688
x15=−15.5792364103872x_{15} = -15.5792364103872
x16=43.9367614714198x_{16} = 43.9367614714198
x17=31.3520917265645x_{17} = 31.3520917265645
x18=53.3695918204908x_{18} = 53.3695918204908
x19=59.6567290035279x_{19} = 59.6567290035279
x20=34.499514921367x_{20} = 34.499514921367
x21=131.931731514843x_{21} = 131.931731514843
x22=47.0813974121542x_{22} = 47.0813974121542
x23=−50.2256516491831x_{23} = -50.2256516491831
x24=94.2265525745684x_{24} = 94.2265525745684
x25=−78.5143405319308x_{25} = -78.5143405319308
x26=−40.7916552312719x_{26} = -40.7916552312719
x27=75.3716854092873x_{27} = 75.3716854092873
x28=69.0860849466452x_{28} = 69.0860849466452
x29=−9.20584014293667x_{29} = -9.20584014293667
x30=65.9431119046552x_{30} = 65.9431119046552
x31=−28.2033610039524x_{31} = -28.2033610039524
x32=−81.6569138240367x_{32} = -81.6569138240367
x33=25.052825280993x_{33} = 25.052825280993
x34=−2.0815759778181x_{34} = -2.0815759778181
x35=−62.8000005565198x_{35} = -62.8000005565198
x36=−94.2265525745684x_{36} = -94.2265525745684
x37=−25.052825280993x_{37} = -25.052825280993
x38=−34.499514921367x_{38} = -34.499514921367
x39=−37.6459603230864x_{39} = -37.6459603230864
x40=28.2033610039524x_{40} = 28.2033610039524
x41=81.6569138240367x_{41} = 81.6569138240367
x42=78.5143405319308x_{42} = 78.5143405319308
x43=56.5132704621986x_{43} = 56.5132704621986
x44=15.5792364103872x_{44} = 15.5792364103872
x45=50.2256516491831x_{45} = 50.2256516491831
x46=97.368830362901x_{46} = 97.368830362901
x47=62.8000005565198x_{47} = 62.8000005565198
x48=87.9418500396598x_{48} = 87.9418500396598
x49=−65.9431119046552x_{49} = -65.9431119046552
x50=40.7916552312719x_{50} = 40.7916552312719
x51=18.7426455847748x_{51} = 18.7426455847748
x52=84.7994143922025x_{52} = 84.7994143922025
x53=−56.5132704621986x_{53} = -56.5132704621986
x54=37.6459603230864x_{54} = 37.6459603230864
x55=72.2289377620154x_{55} = 72.2289377620154
x56=−100.511065295271x_{56} = -100.511065295271
x57=−53.3695918204908x_{57} = -53.3695918204908
x58=−5.94036999057271x_{58} = -5.94036999057271
x59=−31.3520917265645x_{59} = -31.3520917265645
x60=2.0815759778181x_{60} = 2.0815759778181
x61=5.94036999057271x_{61} = 5.94036999057271
x62=12.404445021902x_{62} = 12.404445021902
x63=−97.368830362901x_{63} = -97.368830362901
x64=−21.8996964794928x_{64} = -21.8996964794928
x65=−69.0860849466452x_{65} = -69.0860849466452
x66=−72.2289377620154x_{66} = -72.2289377620154
x67=−75.3716854092873x_{67} = -75.3716854092873
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=0x_{1} = 0

lim⁡x→0−(−sin⁡(x)−2cos⁡(x)x+2sin⁡(x)x23x)=−19\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{2 \sin{\left(x \right)}}{x^{2}}}{3 x}\right) = - \frac{1}{9}
lim⁡x→0+(−sin⁡(x)−2cos⁡(x)x+2sin⁡(x)x23x)=−19\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{2 \sin{\left(x \right)}}{x^{2}}}{3 x}\right) = - \frac{1}{9}
- los límites son iguales, es decir omitimos el punto correspondiente

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
[97.368830362901,∞)\left[97.368830362901, \infty\right)
Convexa en los intervalos
(−∞,−1288.05143523817]\left(-\infty, -1288.05143523817\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(x)3x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{3 x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(x)3x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{3 x}\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 sin(x)/((3*x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(13xsin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{3 x} \sin{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(13xsin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{3 x} \sin{\left(x \right)}}{x}\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:
sin⁡(x)3x=sin⁡(x)3x\frac{\sin{\left(x \right)}}{3 x} = \frac{\sin{\left(x \right)}}{3 x}
- No
sin⁡(x)3x=−sin⁡(x)3x\frac{\sin{\left(x \right)}}{3 x} = - \frac{\sin{\left(x \right)}}{3 x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор