Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • |x|-x |x|-x
  • x^4-4x^2+5
  • -x^3-x-2
  • x^3-4x^2+3x
  • Integral of d{x}:
  • siny/y^3

  • Identical expressions

  • siny/y^ three
  • sinus of y divide by y cubed
  • sinus of y divide by y to the power of three
  • siny/y3
  • siny/y³
  • siny/y to the power of 3
  • siny divide by y^3
Plotting a function graph/ siny/y^3

Graphing y = siny/y^3

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       sin(y)
f(y) = ------
          3  
         y
f(y)=sin⁡(y)y3f{\left(y \right)} = \frac{\sin{\left(y \right)}}{y^{3}} 
f = sin(y)/y^3
The graph of the function
02468-8-6-4-2-1010-500500
The domain of the function
The points at which the function is not precisely defined:
y1=0y_{1} = 0
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis Y at f = 0
so we need to solve the equation:
sin⁡(y)y3=0\frac{\sin{\left(y \right)}}{y^{3}} = 0
Solve this equation
The points of intersection with the axis Y:

Analytical solution
y1=πy_{1} = \pi
Numerical solution
y1=31.4159265358979y_{1} = 31.4159265358979
y2=3.14159265358979y_{2} = 3.14159265358979
y3=−47.1238898038469y_{3} = -47.1238898038469
y4=−12.5663706143592y_{4} = -12.5663706143592
y5=−34.5575191894877y_{5} = -34.5575191894877
y6=−69.1150383789755y_{6} = -69.1150383789755
y7=75.398223686155y_{7} = 75.398223686155
y8=−65.9734457253857y_{8} = -65.9734457253857
y9=−50.2654824574367y_{9} = -50.2654824574367
y10=−56.5486677646163y_{10} = -56.5486677646163
y11=59.6902604182061y_{11} = 59.6902604182061
y12=141.371669411541y_{12} = 141.371669411541
y13=72.2566310325652y_{13} = 72.2566310325652
y14=91.106186954104y_{14} = 91.106186954104
y15=−91.106186954104y_{15} = -91.106186954104
y16=−109.955742875643y_{16} = -109.955742875643
y17=−62.8318530717959y_{17} = -62.8318530717959
y18=−6.28318530717959y_{18} = -6.28318530717959
y19=6.28318530717959y_{19} = 6.28318530717959
y20=62.8318530717959y_{20} = 62.8318530717959
y21=−25.1327412287183y_{21} = -25.1327412287183
y22=94.2477796076938y_{22} = 94.2477796076938
y23=−9.42477796076938y_{23} = -9.42477796076938
y24=−37.6991118430775y_{24} = -37.6991118430775
y25=65.9734457253857y_{25} = 65.9734457253857
y26=−100.530964914873y_{26} = -100.530964914873
y27=−43.9822971502571y_{27} = -43.9822971502571
y28=25.1327412287183y_{28} = 25.1327412287183
y29=21.9911485751286y_{29} = 21.9911485751286
y30=87.9645943005142y_{30} = 87.9645943005142
y31=−40.8407044966673y_{31} = -40.8407044966673
y32=−97.3893722612836y_{32} = -97.3893722612836
y33=43.9822971502571y_{33} = 43.9822971502571
y34=−53.4070751110265y_{34} = -53.4070751110265
y35=97.3893722612836y_{35} = 97.3893722612836
y36=100.530964914873y_{36} = 100.530964914873
y37=−94.2477796076938y_{37} = -94.2477796076938
y38=−31.4159265358979y_{38} = -31.4159265358979
y39=18.8495559215388y_{39} = 18.8495559215388
y40=78.5398163397448y_{40} = 78.5398163397448
y41=−18.8495559215388y_{41} = -18.8495559215388
y42=53.4070751110265y_{42} = 53.4070751110265
y43=47.1238898038469y_{43} = 47.1238898038469
y44=12.5663706143592y_{44} = 12.5663706143592
y45=81.6814089933346y_{45} = 81.6814089933346
y46=34.5575191894877y_{46} = 34.5575191894877
y47=−75.398223686155y_{47} = -75.398223686155
y48=−15.707963267949y_{48} = -15.707963267949
y49=50.2654824574367y_{49} = 50.2654824574367
y50=−81.6814089933346y_{50} = -81.6814089933346
y51=−3.14159265358979y_{51} = -3.14159265358979
y52=−59.6902604182061y_{52} = -59.6902604182061
y53=−28.2743338823081y_{53} = -28.2743338823081
y54=−87.9645943005142y_{54} = -87.9645943005142
y55=9.42477796076938y_{55} = 9.42477796076938
y56=−21.9911485751286y_{56} = -21.9911485751286
y57=56.5486677646163y_{57} = 56.5486677646163
y58=15.707963267949y_{58} = 15.707963267949
y59=84.8230016469244y_{59} = 84.8230016469244
y60=−78.5398163397448y_{60} = -78.5398163397448
y61=37.6991118430775y_{61} = 37.6991118430775
y62=−72.2566310325652y_{62} = -72.2566310325652
y63=−84.8230016469244y_{63} = -84.8230016469244
y64=69.1150383789755y_{64} = 69.1150383789755
y65=28.2743338823081y_{65} = 28.2743338823081
y66=40.8407044966673y_{66} = 40.8407044966673
The points of intersection with the Y axis coordinate
The graph crosses Y axis when y equals 0:
substitute y = 0 to sin(y)/y^3.
sin⁡(0)03\frac{\sin{\left(0 \right)}}{0^{3}}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
Extrema of the function
In order to find the extrema, we need to solve the equation
ddyf(y)=0\frac{d}{d y} f{\left(y \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddyf(y)=\frac{d}{d y} f{\left(y \right)} =
the first derivative
cos⁡(y)y3−3sin⁡(y)y4=0\frac{\cos{\left(y \right)}}{y^{3}} - \frac{3 \sin{\left(y \right)}}{y^{4}} = 0
Solve this equation
The roots of this equation
y1=7.47219265966058y_{1} = 7.47219265966058
y2=−48.6330777853047y_{2} = -48.6330777853047
y3=48.6330777853047y_{3} = 48.6330777853047
y4=86.359073259985y_{4} = 86.359073259985
y5=−13.924969952549y_{5} = -13.924969952549
y6=64.3560674689022y_{6} = 64.3560674689022
y7=−80.0731644462726y_{7} = -80.0731644462726
y8=−51.7784042739041y_{8} = -51.7784042739041
y9=73.7867920572034y_{9} = 73.7867920572034
y10=95.7872667660245y_{10} = 95.7872667660245
y11=45.4872362867621y_{11} = 45.4872362867621
y12=−108.357267428671y_{12} = -108.357267428671
y13=−26.591193287969y_{13} = -26.591193287969
y14=−23.4346216921802y_{14} = -23.4346216921802
y15=−20.2734415170608y_{15} = -20.2734415170608
y16=76.930043294192y_{16} = 76.930043294192
y17=32.8957773192946y_{17} = 32.8957773192946
y18=39.1935138550425y_{18} = 39.1935138550425
y19=−95.7872667660245y_{19} = -95.7872667660245
y20=−76.930043294192y_{20} = -76.930043294192
y21=−42.3407653325706y_{21} = -42.3407653325706
y22=−73.7867920572034y_{22} = -73.7867920572034
y23=80.0731644462726y_{23} = 80.0731644462726
y24=−61.2120859995403y_{24} = -61.2120859995403
y25=83.2161702400252y_{25} = 83.2161702400252
y26=−83.2161702400252y_{26} = -83.2161702400252
y27=98.9298533613919y_{27} = 98.9298533613919
y28=67.4998267230665y_{28} = 67.4998267230665
y29=−98.9298533613919y_{29} = -98.9298533613919
y30=−186.908713650658y_{30} = -186.908713650658
y31=10.7227710626892y_{31} = 10.7227710626892
y32=89.5018843244389y_{32} = 89.5018843244389
y33=17.1051395364267y_{33} = 17.1051395364267
y34=36.0452782424582y_{34} = 36.0452782424582
y35=−86.359073259985y_{35} = -86.359073259985
y36=−29.7446115259422y_{36} = -29.7446115259422
y37=23.4346216921802y_{37} = 23.4346216921802
y38=92.6446127847888y_{38} = 92.6446127847888
y39=4.07814976485137y_{39} = 4.07814976485137
y40=54.9233040395155y_{40} = 54.9233040395155
y41=61.2120859995403y_{41} = 61.2120859995403
y42=−7.47219265966058y_{42} = -7.47219265966058
y43=29.7446115259422y_{43} = 29.7446115259422
y44=−39.1935138550425y_{44} = -39.1935138550425
y45=−58.0678462801751y_{45} = -58.0678462801751
y46=−70.6433933906731y_{46} = -70.6433933906731
y47=3215.41914794509y_{47} = 3215.41914794509
y48=58.0678462801751y_{48} = 58.0678462801751
y49=20.2734415170608y_{49} = 20.2734415170608
y50=−67.4998267230665y_{50} = -67.4998267230665
y51=42.3407653325706y_{51} = 42.3407653325706
y52=26.591193287969y_{52} = 26.591193287969
y53=−17.1051395364267y_{53} = -17.1051395364267
y54=−54.9233040395155y_{54} = -54.9233040395155
y55=70.6433933906731y_{55} = 70.6433933906731
y56=−32.8957773192946y_{56} = -32.8957773192946
y57=−4.07814976485137y_{57} = -4.07814976485137
y58=13.924969952549y_{58} = 13.924969952549
y59=−64.3560674689022y_{59} = -64.3560674689022
y60=−89.5018843244389y_{60} = -89.5018843244389
y61=51.7784042739041y_{61} = 51.7784042739041
y62=−45.4872362867621y_{62} = -45.4872362867621
y63=−92.6446127847888y_{63} = -92.6446127847888
y64=−10.7227710626892y_{64} = -10.7227710626892
y65=−36.0452782424582y_{65} = -36.0452782424582
The values of the extrema at the points:
(7.472192659660579, 0.00222435242575847)

(-48.63307778530466, -8.67720806398467e-6)

(48.63307778530466, -8.67720806398467e-6)

(86.359073259985, -1.55172297524868e-6)

(-13.92496995254897, 0.000362047304723488)

(64.35606746890217, 3.7476597286644e-6)

(-80.07316444627257, -1.94641047170913e-6)

(-51.77840427390411, 7.1916125507828e-6)

(73.78679205720341, -2.48716990126569e-6)

(95.78726676602449, 1.1372704641271e-6)

(45.48723628676209, 1.06020259212848e-5)

(-108.35726742867121, 7.85704936475449e-7)

(-26.59119328796898, 5.28494009082656e-5)

(-23.4346216921802, -7.70719574291125e-5)

(-20.27344151706078, 0.000118717289027919)

(76.93004329419203, 2.19473504875366e-6)

(32.89577731929462, 2.79757033726946e-5)

(39.193513855042454, 1.65610881918558e-5)

(-95.78726676602449, 1.1372704641271e-6)

(-76.93004329419203, 2.19473504875366e-6)

(-42.34076533257061, -1.31412441116291e-5)

(-73.78679205720341, -2.48716990126569e-6)

(80.07316444627257, -1.94641047170913e-6)

(-61.21208599954033, -4.35479288166118e-6)

(83.21617024002518, 1.73418247352317e-6)

(-83.21617024002518, 1.73418247352317e-6)

(98.9298533613919, -1.03232943885669e-6)

(67.49982672306646, -3.24835521431348e-6)

(-98.9298533613919, -1.03232943885669e-6)

(-186.90871365065752, -1.53128285331065e-7)

(10.722771062689203, -0.00078111312666525)

(89.50188432443886, 1.39398991793862e-6)

(17.105139536426744, -0.000196807350078387)

(36.04527824245817, -2.12792275158681e-5)

(-86.359073259985, -1.55172297524868e-6)

(-29.744611525942226, -3.78074454700618e-5)

(23.4346216921802, -7.70719574291125e-5)

(92.64461278478876, -1.25693237122389e-6)

(4.078149764851372, -0.0118764951343876)

(54.92330403951548, -6.02674942320184e-6)

(61.21208599954033, -4.35479288166118e-6)

(-7.472192659660579, 0.00222435242575847)

(29.744611525942226, -3.78074454700618e-5)

(-39.193513855042454, 1.65610881918558e-5)

(-58.067846280175104, 5.1005149012716e-6)

(-70.64339339067311, 2.83396240646379e-6)

(3215.41914794509, -3.00806370783767e-11)

(58.067846280175104, 5.1005149012716e-6)

(20.27344151706078, 0.000118717289027919)

(-67.49982672306646, -3.24835521431348e-6)

(42.34076533257061, -1.31412441116291e-5)

(26.59119328796898, 5.28494009082656e-5)

(-17.105139536426744, -0.000196807350078387)

(-54.92330403951548, -6.02674942320184e-6)

(70.64339339067311, 2.83396240646379e-6)

(-32.89577731929462, 2.79757033726946e-5)

(-4.078149764851372, -0.0118764951343876)

(13.92496995254897, 0.000362047304723488)

(-64.35606746890217, 3.7476597286644e-6)

(-89.50188432443886, 1.39398991793862e-6)

(51.77840427390411, 7.1916125507828e-6)

(-45.48723628676209, 1.06020259212848e-5)

(-92.64461278478876, -1.25693237122389e-6)

(-10.722771062689203, -0.00078111312666525)

(-36.04527824245817, -2.12792275158681e-5)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
y1=−48.6330777853047y_{1} = -48.6330777853047
y2=48.6330777853047y_{2} = 48.6330777853047
y3=86.359073259985y_{3} = 86.359073259985
y4=−80.0731644462726y_{4} = -80.0731644462726
y5=73.7867920572034y_{5} = 73.7867920572034
y6=−23.4346216921802y_{6} = -23.4346216921802
y7=−42.3407653325706y_{7} = -42.3407653325706
y8=−73.7867920572034y_{8} = -73.7867920572034
y9=80.0731644462726y_{9} = 80.0731644462726
y10=−61.2120859995403y_{10} = -61.2120859995403
y11=98.9298533613919y_{11} = 98.9298533613919
y12=67.4998267230665y_{12} = 67.4998267230665
y13=−98.9298533613919y_{13} = -98.9298533613919
y14=−186.908713650658y_{14} = -186.908713650658
y15=10.7227710626892y_{15} = 10.7227710626892
y16=17.1051395364267y_{16} = 17.1051395364267
y17=36.0452782424582y_{17} = 36.0452782424582
y18=−86.359073259985y_{18} = -86.359073259985
y19=−29.7446115259422y_{19} = -29.7446115259422
y20=23.4346216921802y_{20} = 23.4346216921802
y21=92.6446127847888y_{21} = 92.6446127847888
y22=4.07814976485137y_{22} = 4.07814976485137
y23=54.9233040395155y_{23} = 54.9233040395155
y24=61.2120859995403y_{24} = 61.2120859995403
y25=29.7446115259422y_{25} = 29.7446115259422
y26=3215.41914794509y_{26} = 3215.41914794509
y27=−67.4998267230665y_{27} = -67.4998267230665
y28=42.3407653325706y_{28} = 42.3407653325706
y29=−17.1051395364267y_{29} = -17.1051395364267
y30=−54.9233040395155y_{30} = -54.9233040395155
y31=−4.07814976485137y_{31} = -4.07814976485137
y32=−92.6446127847888y_{32} = -92.6446127847888
y33=−10.7227710626892y_{33} = -10.7227710626892
y34=−36.0452782424582y_{34} = -36.0452782424582
Maxima of the function at points:
y34=7.47219265966058y_{34} = 7.47219265966058
y34=−13.924969952549y_{34} = -13.924969952549
y34=64.3560674689022y_{34} = 64.3560674689022
y34=−51.7784042739041y_{34} = -51.7784042739041
y34=95.7872667660245y_{34} = 95.7872667660245
y34=45.4872362867621y_{34} = 45.4872362867621
y34=−108.357267428671y_{34} = -108.357267428671
y34=−26.591193287969y_{34} = -26.591193287969
y34=−20.2734415170608y_{34} = -20.2734415170608
y34=76.930043294192y_{34} = 76.930043294192
y34=32.8957773192946y_{34} = 32.8957773192946
y34=39.1935138550425y_{34} = 39.1935138550425
y34=−95.7872667660245y_{34} = -95.7872667660245
y34=−76.930043294192y_{34} = -76.930043294192
y34=83.2161702400252y_{34} = 83.2161702400252
y34=−83.2161702400252y_{34} = -83.2161702400252
y34=89.5018843244389y_{34} = 89.5018843244389
y34=−7.47219265966058y_{34} = -7.47219265966058
y34=−39.1935138550425y_{34} = -39.1935138550425
y34=−58.0678462801751y_{34} = -58.0678462801751
y34=−70.6433933906731y_{34} = -70.6433933906731
y34=58.0678462801751y_{34} = 58.0678462801751
y34=20.2734415170608y_{34} = 20.2734415170608
y34=26.591193287969y_{34} = 26.591193287969
y34=70.6433933906731y_{34} = 70.6433933906731
y34=−32.8957773192946y_{34} = -32.8957773192946
y34=13.924969952549y_{34} = 13.924969952549
y34=−64.3560674689022y_{34} = -64.3560674689022
y34=−89.5018843244389y_{34} = -89.5018843244389
y34=51.7784042739041y_{34} = 51.7784042739041
y34=−45.4872362867621y_{34} = -45.4872362867621
Decreasing at intervals
[3215.41914794509,∞)\left[3215.41914794509, \infty\right)
Increasing at intervals
(−∞,−186.908713650658]\left(-\infty, -186.908713650658\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dy2f(y)=0\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dy2f(y)=\frac{d^{2}}{d y^{2}} f{\left(y \right)} =
the second derivative
−sin⁡(y)−6cos⁡(y)y+12sin⁡(y)y2y3=0\frac{- \sin{\left(y \right)} - \frac{6 \cos{\left(y \right)}}{y} + \frac{12 \sin{\left(y \right)}}{y^{2}}}{y^{3}} = 0
Solve this equation
The roots of this equation
y1=43.8454539221714y_{1} = 43.8454539221714
y2=50.1458319794503y_{2} = 50.1458319794503
y3=15.3164244547999y_{3} = 15.3164244547999
y4=−141.329215347444y_{4} = -141.329215347444
y5=97.3277248932537y_{5} = 97.3277248932537
y6=21.7148754289157y_{6} = 21.7148754289157
y7=94.1840745935286y_{7} = 94.1840745935286
y8=−72.1734981126022y_{8} = -72.1734981126022
y9=87.8963320992957y_{9} = 87.8963320992957
y10=−81.6078867352652y_{10} = -81.6078867352652
y11=373.833475850202y_{11} = 373.833475850202
y12=−12.0699013399661y_{12} = -12.0699013399661
y13=46.996220714232y_{13} = 46.996220714232
y14=8.74140008690353y_{14} = 8.74140008690353
y15=72.1734981126022y_{15} = 72.1734981126022
y16=−59.5895718893518y_{16} = -59.5895718893518
y17=−78.4633475726977y_{17} = -78.4633475726977
y18=−15.3164244547999y_{18} = -15.3164244547999
y19=34.3830180083391y_{19} = 34.3830180083391
y20=37.5392815729796y_{20} = 37.5392815729796
y21=−97.3277248932537y_{21} = -97.3277248932537
y22=24.8917150033836y_{22} = 24.8917150033836
y23=18.5257597776303y_{23} = 18.5257597776303
y24=84.7522070698598y_{24} = 84.7522070698598
y25=−46.996220714232y_{25} = -46.996220714232
y26=91.0402820892519y_{26} = 91.0402820892519
y27=75.31856211974y_{27} = 75.31856211974
y28=5.15216592622293y_{28} = 5.15216592622293
y29=−62.7362147088964y_{29} = -62.7362147088964
y30=116.187287429474y_{30} = 116.187287429474
y31=69.028117387504y_{31} = 69.028117387504
y32=78.4633475726977y_{32} = 78.4633475726977
y33=−28.0605201580983y_{33} = -28.0605201580983
y34=−40.6932614780489y_{34} = -40.6932614780489
y35=40.6932614780489y_{35} = 40.6932614780489
y36=31.223771021093y_{36} = 31.223771021093
y37=−53.2944935242974y_{37} = -53.2944935242974
y38=56.4423649364807y_{38} = 56.4423649364807
y39=−21.7148754289157y_{39} = -21.7148754289157
y40=−37.5392815729796y_{40} = -37.5392815729796
y41=131.901402932105y_{41} = 131.901402932105
y42=100.47124635336y_{42} = 100.47124635336
y43=81.6078867352652y_{43} = 81.6078867352652
y44=−34.3830180083391y_{44} = -34.3830180083391
y45=62.7362147088964y_{45} = 62.7362147088964
y46=59.5895718893518y_{46} = 59.5895718893518
y47=65.8823744655118y_{47} = 65.8823744655118
y48=−75.31856211974y_{48} = -75.31856211974
y49=−69.028117387504y_{49} = -69.028117387504
y50=−84.7522070698598y_{50} = -84.7522070698598
y51=−56.4423649364807y_{51} = -56.4423649364807
y52=−31.223771021093y_{52} = -31.223771021093
y53=−18.5257597776303y_{53} = -18.5257597776303
y54=−50.1458319794503y_{54} = -50.1458319794503
y55=−94.1840745935286y_{55} = -94.1840745935286
y56=−24.8917150033836y_{56} = -24.8917150033836
y57=−113.044258982077y_{57} = -113.044258982077
y58=−5.15216592622293y_{58} = -5.15216592622293
y59=−65.8823744655118y_{59} = -65.8823744655118
y60=−91.0402820892519y_{60} = -91.0402820892519
y61=−43.8454539221714y_{61} = -43.8454539221714
y62=−100.47124635336y_{62} = -100.47124635336
y63=−87.8963320992957y_{63} = -87.8963320992957
y64=12.0699013399661y_{64} = 12.0699013399661
y65=−8.74140008690353y_{65} = -8.74140008690353
y66=28.0605201580983y_{66} = 28.0605201580983
y67=53.2944935242974y_{67} = 53.2944935242974
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
y1=0y_{1} = 0

lim⁡y→0−(−sin⁡(y)−6cos⁡(y)y+12sin⁡(y)y2y3)=∞\lim_{y \to 0^-}\left(\frac{- \sin{\left(y \right)} - \frac{6 \cos{\left(y \right)}}{y} + \frac{12 \sin{\left(y \right)}}{y^{2}}}{y^{3}}\right) = \infty
lim⁡y→0+(−sin⁡(y)−6cos⁡(y)y+12sin⁡(y)y2y3)=∞\lim_{y \to 0^+}\left(\frac{- \sin{\left(y \right)} - \frac{6 \cos{\left(y \right)}}{y} + \frac{12 \sin{\left(y \right)}}{y^{2}}}{y^{3}}\right) = \infty
- limits are equal, then skip the corresponding point

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[373.833475850202,∞)\left[373.833475850202, \infty\right)
Convex at the intervals
(−∞,−113.044258982077]\left(-\infty, -113.044258982077\right]
Vertical asymptotes
Have:
y1=0y_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at y->+oo and y->-oo
lim⁡y→−∞(sin⁡(y)y3)=0\lim_{y \to -\infty}\left(\frac{\sin{\left(y \right)}}{y^{3}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡y→∞(sin⁡(y)y3)=0\lim_{y \to \infty}\left(\frac{\sin{\left(y \right)}}{y^{3}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(y)/y^3, divided by y at y->+oo and y ->-oo
lim⁡y→−∞(sin⁡(y)yy3)=0\lim_{y \to -\infty}\left(\frac{\sin{\left(y \right)}}{y y^{3}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡y→∞(sin⁡(y)yy3)=0\lim_{y \to \infty}\left(\frac{\sin{\left(y \right)}}{y y^{3}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-y) и f = -f(-y).
So, check:
sin⁡(y)y3=sin⁡(y)y3\frac{\sin{\left(y \right)}}{y^{3}} = \frac{\sin{\left(y \right)}}{y^{3}}
- No
sin⁡(y)y3=−sin⁡(y)y3\frac{\sin{\left(y \right)}}{y^{3}} = - \frac{\sin{\left(y \right)}}{y^{3}}
- No
so, the function
not is
neither even, nor odd
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