Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • x^6+x^3-2
  • x-3/4-x
  • (x-3)²
  • x^3-12x-7
  • Limit of the function:
  • sin(x)/x^4 sin(x)/x^4

  • Identical expressions

  • sin(x)/x^ four
  • sinus of (x) divide by x to the power of 4
  • sinus of (x) divide by x to the power of four
  • sin(x)/x4
  • sinx/x4
  • sin(x)/x⁴
  • sinx/x^4
  • sin(x) divide by x^4

  • Similar expressions

  • sinx/x^4
Plotting a function graph/ sin(x)/x^4

Graphing y = sin(x)/x^4

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       sin(x)
f(x) = ------
          4  
         x
f(x)=sin⁡(x)x4f{\left(x \right)} = \frac{\sin{\left(x \right)}}{x^{4}} 
f = sin(x)/x^4
The graph of the function
02468-8-6-4-2-1010-2000020000
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
sin⁡(x)x4=0\frac{\sin{\left(x \right)}}{x^{4}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=πx_{1} = \pi
Numerical solution
x1=72.2566310325652x_{1} = 72.2566310325652
x2=−103.672557568463x_{2} = -103.672557568463
x3=−59.6902604182061x_{3} = -59.6902604182061
x4=3.14159265358979x_{4} = 3.14159265358979
x5=−43.9822971502571x_{5} = -43.9822971502571
x6=81.6814089933346x_{6} = 81.6814089933346
x7=−100.530964914873x_{7} = -100.530964914873
x8=28.2743338823081x_{8} = 28.2743338823081
x9=65.9734457253857x_{9} = 65.9734457253857
x10=−31.4159265358979x_{10} = -31.4159265358979
x11=−9.42477796076938x_{11} = -9.42477796076938
x12=40.8407044966673x_{12} = 40.8407044966673
x13=56.5486677646163x_{13} = 56.5486677646163
x14=−56.5486677646163x_{14} = -56.5486677646163
x15=12.5663706143592x_{15} = 12.5663706143592
x16=43.9822971502571x_{16} = 43.9822971502571
x17=100.530964914873x_{17} = 100.530964914873
x18=−3.14159265358979x_{18} = -3.14159265358979
x19=−15.707963267949x_{19} = -15.707963267949
x20=59.6902604182061x_{20} = 59.6902604182061
x21=6.28318530717959x_{21} = 6.28318530717959
x22=9.42477796076938x_{22} = 9.42477796076938
x23=−53.4070751110265x_{23} = -53.4070751110265
x24=−47.1238898038469x_{24} = -47.1238898038469
x25=−87.9645943005142x_{25} = -87.9645943005142
x26=69.1150383789755x_{26} = 69.1150383789755
x27=21.9911485751286x_{27} = 21.9911485751286
x28=87.9645943005142x_{28} = 87.9645943005142
x29=18.8495559215388x_{29} = 18.8495559215388
x30=−84.8230016469244x_{30} = -84.8230016469244
x31=−72.2566310325652x_{31} = -72.2566310325652
x32=25.1327412287183x_{32} = 25.1327412287183
x33=37.6991118430775x_{33} = 37.6991118430775
x34=−25.1327412287183x_{34} = -25.1327412287183
x35=50.2654824574367x_{35} = 50.2654824574367
x36=34.5575191894877x_{36} = 34.5575191894877
x37=−6.28318530717959x_{37} = -6.28318530717959
x38=−65.9734457253857x_{38} = -65.9734457253857
x39=−21.9911485751286x_{39} = -21.9911485751286
x40=−62.8318530717959x_{40} = -62.8318530717959
x41=75.398223686155x_{41} = 75.398223686155
x42=84.8230016469244x_{42} = 84.8230016469244
x43=53.4070751110265x_{43} = 53.4070751110265
x44=15.707963267949x_{44} = 15.707963267949
x45=−28.2743338823081x_{45} = -28.2743338823081
x46=−91.106186954104x_{46} = -91.106186954104
x47=47.1238898038469x_{47} = 47.1238898038469
x48=97.3893722612836x_{48} = 97.3893722612836
x49=−69.1150383789755x_{49} = -69.1150383789755
x50=94.2477796076938x_{50} = 94.2477796076938
x51=−18.8495559215388x_{51} = -18.8495559215388
x52=−50.2654824574367x_{52} = -50.2654824574367
x53=−37.6991118430775x_{53} = -37.6991118430775
x54=−81.6814089933346x_{54} = -81.6814089933346
x55=62.8318530717959x_{55} = 62.8318530717959
x56=78.5398163397448x_{56} = 78.5398163397448
x57=31.4159265358979x_{57} = 31.4159265358979
x58=−78.5398163397448x_{58} = -78.5398163397448
x59=−40.8407044966673x_{59} = -40.8407044966673
x60=−97.3893722612836x_{60} = -97.3893722612836
x61=−75.398223686155x_{61} = -75.398223686155
x62=91.106186954104x_{62} = 91.106186954104
x63=−12.5663706143592x_{63} = -12.5663706143592
x64=−94.2477796076938x_{64} = -94.2477796076938
x65=−34.5575191894877x_{65} = -34.5575191894877
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)/x^4.
sin⁡(0)04\frac{\sin{\left(0 \right)}}{0^{4}}
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
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
cos⁡(x)x4−4sin⁡(x)x5=0\frac{\cos{\left(x \right)}}{x^{4}} - \frac{4 \sin{\left(x \right)}}{x^{5}} = 0
Solve this equation
The roots of this equation
x1=58.0506675115744x_{1} = 58.0506675115744
x2=−26.554025372648x_{2} = -26.554025372648
x3=−67.4850389217693x_{3} = -67.4850389217693
x4=−86.3475066265478x_{4} = -86.3475066265478
x5=102.062589651767x_{5} = 102.062589651767
x6=−3.91643536817833x_{6} = -3.91643536817833
x7=61.1957856169513x_{7} = 61.1957856169513
x8=−95.7768364391699x_{8} = -95.7768364391699
x9=−58.0506675115744x_{9} = -58.0506675115744
x10=−83.2041677899554x_{10} = -83.2041677899554
x11=−23.3925885027245x_{11} = -23.3925885027245
x12=39.1681371651634x_{12} = 39.1681371651634
x13=7.35592702313142x_{13} = 7.35592702313142
x14=73.7732602110395x_{14} = 73.7732602110395
x15=45.4653403180975x_{15} = 45.4653403180975
x16=−48.6125878655427x_{16} = -48.6125878655427
x17=−80.0606920801604x_{17} = -80.0606920801604
x18=42.3172567562625x_{18} = 42.3172567562625
x19=−17.0483006782016x_{19} = -17.0483006782016
x20=−98.9197537891963x_{20} = -98.9197537891963
x21=76.9170627514565x_{21} = 76.9170627514565
x22=20.2250979158179x_{22} = 20.2250979158179
x23=−73.7732602110395x_{23} = -73.7732602110395
x24=−70.6292613872399x_{24} = -70.6292613872399
x25=95.7768364391699x_{25} = 95.7768364391699
x26=98.9197537891963x_{26} = 98.9197537891963
x27=120.918249020695x_{27} = 120.918249020695
x28=51.7591510649689x_{28} = 51.7591510649689
x29=70.6292613872399x_{29} = 70.6292613872399
x30=64.3405601260236x_{30} = 64.3405601260236
x31=−45.4653403180975x_{31} = -45.4653403180975
x32=89.4907229874848x_{32} = 89.4907229874848
x33=−13.856126658747x_{33} = -13.856126658747
x34=26.554025372648x_{34} = 26.554025372648
x35=54.905147004153x_{35} = 54.905147004153
x36=−39.1681371651634x_{36} = -39.1681371651634
x37=13.856126658747x_{37} = 13.856126658747
x38=−10.6358514209244x_{38} = -10.6358514209244
x39=67.4850389217693x_{39} = 67.4850389217693
x40=252.882392303412x_{40} = 252.882392303412
x41=80.0606920801604x_{41} = 80.0606920801604
x42=−89.4907229874848x_{42} = -89.4907229874848
x43=86.3475066265478x_{43} = 86.3475066265478
x44=−76.9170627514565x_{44} = -76.9170627514565
x45=−32.8656107569429x_{45} = -32.8656107569429
x46=−61.1957856169513x_{46} = -61.1957856169513
x47=−36.0177122696989x_{47} = -36.0177122696989
x48=−64.3405601260236x_{48} = -64.3405601260236
x49=−20.2250979158179x_{49} = -20.2250979158179
x50=−142.914484277841x_{50} = -142.914484277841
x51=10.6358514209244x_{51} = 10.6358514209244
x52=−922.053105710894x_{52} = -922.053105710894
x53=48.6125878655427x_{53} = 48.6125878655427
x54=29.7113059713865x_{54} = 29.7113059713865
x55=23.3925885027245x_{55} = 23.3925885027245
x56=−92.6338293197393x_{56} = -92.6338293197393
x57=−7.35592702313142x_{57} = -7.35592702313142
x58=92.6338293197393x_{58} = 92.6338293197393
x59=−42.3172567562625x_{59} = -42.3172567562625
x60=17.0483006782016x_{60} = 17.0483006782016
x61=−51.7591510649689x_{61} = -51.7591510649689
x62=32.8656107569429x_{62} = 32.8656107569429
x63=83.2041677899554x_{63} = 83.2041677899554
x64=−54.905147004153x_{64} = -54.905147004153
x65=36.0177122696989x_{65} = 36.0177122696989
x66=−29.7113059713865x_{66} = -29.7113059713865
x67=3.91643536817833x_{67} = 3.91643536817833
The values of the extrema at the points:
(58.05066751157438, 8.78501616937435e-8)

(-26.554025372648002, -1.98886935969342e-6)

(-67.48503892176933, 4.81291775980568e-8)

(-86.3475066265478, 1.7969471563081e-8)

(102.06258965176721, 9.20874372329637e-9)

(-3.916435368178333, 0.00297363898119923)

(61.195785616951255, -7.11521729092172e-8)

(-95.77683643916993, -1.1873523810379e-8)

(-58.05066751157438, -8.78501616937435e-8)

(-83.20416778995545, -2.08409922801071e-8)

(-23.392588502724486, 3.29176440749978e-6)

(39.168137165163394, 4.22683574600292e-7)

(7.355927023131424, 0.000300053589186415)

(73.77326021103951, -3.37106134268981e-8)

(45.46534031809747, 2.33133077593193e-7)

(-48.612587865542714, 1.78459548054671e-7)

(-80.06069208016042, 2.43097935138504e-8)

(42.31725675626252, -3.10454831057143e-7)

(-17.0483006782016, 1.15249508377501e-5)

(-98.91975378919633, 1.04354965234194e-8)

(76.91706275145651, 2.85313794171889e-8)

(20.22509791581794, 5.86280903576885e-6)

(-73.77326021103951, 3.37106134268981e-8)

(-70.62926138723986, -4.01204672332662e-8)

(95.77683643916993, 1.1873523810379e-8)

(98.91975378919633, -1.04354965234194e-8)

(120.91824902069463, 4.67514523669302e-9)

(51.759151064968904, 1.38917960418904e-7)

(70.62926138723986, 4.01204672332662e-8)

(64.34056012602356, 5.82402180776872e-8)

(-45.46534031809747, -2.33133077593193e-7)

(89.49072298748483, 1.55759497424426e-8)

(-13.856126658747035, -2.60645897698916e-5)

(26.554025372648002, 1.98886935969342e-6)

(54.90514700415301, -1.09748424351353e-7)

(-39.168137165163394, -4.22683574600292e-7)

(13.856126658747035, 2.60645897698916e-5)

(-10.635851420924443, 7.31449268885113e-5)

(67.48503892176933, -4.81291775980568e-8)

(252.88239230341162, 2.44495748568497e-10)

(80.06069208016042, -2.43097935138504e-8)

(-89.49072298748483, -1.55759497424426e-8)

(86.3475066265478, -1.7969471563081e-8)

(-76.91706275145651, -2.85313794171889e-8)

(-32.86561075694293, -8.50824929781849e-7)

(-61.195785616951255, 7.11521729092172e-8)

(-36.017712269698876, 5.90573138490323e-7)

(-64.34056012602356, -5.82402180776872e-8)

(-20.22509791581794, -5.86280903576885e-6)

(-142.9144842778405, 2.39621053902668e-9)

(10.635851420924443, -7.31449268885113e-5)

(-922.0531057108941, 1.38347773259507e-12)

(48.612587865542714, -1.78459548054671e-7)

(29.711305971386484, -1.27178147734334e-6)

(23.392588502724486, -3.29176440749978e-6)

(-92.63382931973935, 1.35680370015572e-8)

(-7.355927023131424, -0.000300053589186415)

(92.63382931973935, -1.35680370015572e-8)

(-42.31725675626252, 3.10454831057143e-7)

(17.0483006782016, -1.15249508377501e-5)

(-51.759151064968904, -1.38917960418904e-7)

(32.86561075694293, 8.50824929781849e-7)

(83.20416778995545, 2.08409922801071e-8)

(-54.90514700415301, 1.09748424351353e-7)

(36.017712269698876, -5.90573138490323e-7)

(-29.711305971386484, 1.27178147734334e-6)

(3.916435368178333, -0.00297363898119923)


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:
x1=−26.554025372648x_{1} = -26.554025372648
x2=61.1957856169513x_{2} = 61.1957856169513
x3=−95.7768364391699x_{3} = -95.7768364391699
x4=−58.0506675115744x_{4} = -58.0506675115744
x5=−83.2041677899554x_{5} = -83.2041677899554
x6=73.7732602110395x_{6} = 73.7732602110395
x7=42.3172567562625x_{7} = 42.3172567562625
x8=−70.6292613872399x_{8} = -70.6292613872399
x9=98.9197537891963x_{9} = 98.9197537891963
x10=−45.4653403180975x_{10} = -45.4653403180975
x11=−13.856126658747x_{11} = -13.856126658747
x12=54.905147004153x_{12} = 54.905147004153
x13=−39.1681371651634x_{13} = -39.1681371651634
x14=67.4850389217693x_{14} = 67.4850389217693
x15=80.0606920801604x_{15} = 80.0606920801604
x16=−89.4907229874848x_{16} = -89.4907229874848
x17=86.3475066265478x_{17} = 86.3475066265478
x18=−76.9170627514565x_{18} = -76.9170627514565
x19=−32.8656107569429x_{19} = -32.8656107569429
x20=−64.3405601260236x_{20} = -64.3405601260236
x21=−20.2250979158179x_{21} = -20.2250979158179
x22=10.6358514209244x_{22} = 10.6358514209244
x23=48.6125878655427x_{23} = 48.6125878655427
x24=29.7113059713865x_{24} = 29.7113059713865
x25=23.3925885027245x_{25} = 23.3925885027245
x26=−7.35592702313142x_{26} = -7.35592702313142
x27=92.6338293197393x_{27} = 92.6338293197393
x28=17.0483006782016x_{28} = 17.0483006782016
x29=−51.7591510649689x_{29} = -51.7591510649689
x30=36.0177122696989x_{30} = 36.0177122696989
x31=3.91643536817833x_{31} = 3.91643536817833
Maxima of the function at points:
x31=58.0506675115744x_{31} = 58.0506675115744
x31=−67.4850389217693x_{31} = -67.4850389217693
x31=−86.3475066265478x_{31} = -86.3475066265478
x31=102.062589651767x_{31} = 102.062589651767
x31=−3.91643536817833x_{31} = -3.91643536817833
x31=−23.3925885027245x_{31} = -23.3925885027245
x31=39.1681371651634x_{31} = 39.1681371651634
x31=7.35592702313142x_{31} = 7.35592702313142
x31=45.4653403180975x_{31} = 45.4653403180975
x31=−48.6125878655427x_{31} = -48.6125878655427
x31=−80.0606920801604x_{31} = -80.0606920801604
x31=−17.0483006782016x_{31} = -17.0483006782016
x31=−98.9197537891963x_{31} = -98.9197537891963
x31=76.9170627514565x_{31} = 76.9170627514565
x31=20.2250979158179x_{31} = 20.2250979158179
x31=−73.7732602110395x_{31} = -73.7732602110395
x31=95.7768364391699x_{31} = 95.7768364391699
x31=120.918249020695x_{31} = 120.918249020695
x31=51.7591510649689x_{31} = 51.7591510649689
x31=70.6292613872399x_{31} = 70.6292613872399
x31=64.3405601260236x_{31} = 64.3405601260236
x31=89.4907229874848x_{31} = 89.4907229874848
x31=26.554025372648x_{31} = 26.554025372648
x31=13.856126658747x_{31} = 13.856126658747
x31=−10.6358514209244x_{31} = -10.6358514209244
x31=252.882392303412x_{31} = 252.882392303412
x31=−61.1957856169513x_{31} = -61.1957856169513
x31=−36.0177122696989x_{31} = -36.0177122696989
x31=−142.914484277841x_{31} = -142.914484277841
x31=−922.053105710894x_{31} = -922.053105710894
x31=−92.6338293197393x_{31} = -92.6338293197393
x31=−42.3172567562625x_{31} = -42.3172567562625
x31=32.8656107569429x_{31} = 32.8656107569429
x31=83.2041677899554x_{31} = 83.2041677899554
x31=−54.905147004153x_{31} = -54.905147004153
x31=−29.7113059713865x_{31} = -29.7113059713865
Decreasing at intervals
[98.9197537891963,∞)\left[98.9197537891963, \infty\right)
Increasing at intervals
(−∞,−95.7768364391699]\left(-\infty, -95.7768364391699\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−sin⁡(x)−8cos⁡(x)x+20sin⁡(x)x2x4=0\frac{- \sin{\left(x \right)} - \frac{8 \cos{\left(x \right)}}{x} + \frac{20 \sin{\left(x \right)}}{x^{2}}}{x^{4}} = 0
Solve this equation
The roots of this equation
x1=40.6440368534098x_{1} = 40.6440368534098
x2=−56.4069014275054x_{2} = -56.4069014275054
x3=65.8519989258876x_{3} = 65.8519989258876
x4=−87.8735702250483x_{4} = -87.8735702250483
x5=91.0183067721586x_{5} = 91.0183067721586
x6=94.1628332382898x_{6} = 94.1628332382898
x7=−46.9536140378669x_{7} = -46.9536140378669
x8=−91.0183067721586x_{8} = -91.0183067721586
x9=34.3247248305535x_{9} = 34.3247248305535
x10=56.4069014275054x_{10} = 56.4069014275054
x11=−50.1059069640945x_{11} = -50.1059069640945
x12=−43.7997777505446x_{12} = -43.7997777505446
x13=78.4378470253956x_{13} = 78.4378470253956
x14=−84.7286001727485x_{14} = -84.7286001727485
x15=−18.4171002799742x_{15} = -18.4171002799742
x16=62.7043139073477x_{16} = 62.7043139073477
x17=−4.78878505737886x_{17} = -4.78878505737886
x18=−505.780600175478x_{18} = -505.780600175478
x19=−97.307170013851x_{19} = -97.307170013851
x20=−27.989021578396x_{20} = -27.989021578396
x21=75.2919958831875x_{21} = 75.2919958831875
x22=−392.678708985305x_{22} = -392.678708985305
x23=−75.2919958831875x_{23} = -75.2919958831875
x24=43.7997777505446x_{24} = 43.7997777505446
x25=−65.8519989258876x_{25} = -65.8519989258876
x26=84.7286001727485x_{26} = 84.7286001727485
x27=−31.1595522204503x_{27} = -31.1595522204503
x28=53.2569316655778x_{28} = 53.2569316655778
x29=100.451334931507x_{29} = 100.451334931507
x30=97.307170013851x_{30} = 97.307170013851
x31=4.78878505737886x_{31} = 4.78878505737886
x32=−11.9023123123852x_{32} = -11.9023123123852
x33=21.6223139880194x_{33} = 21.6223139880194
x34=−62.7043139073477x_{34} = -62.7043139073477
x35=−609.455848379961x_{35} = -609.455848379961
x36=18.4171002799742x_{36} = 18.4171002799742
x37=24.8110526666222x_{37} = 24.8110526666222
x38=−81.5833695706461x_{38} = -81.5833695706461
x39=−21.6223139880194x_{39} = -21.6223139880194
x40=−68.9991276649867x_{40} = -68.9991276649867
x41=72.145773078307x_{41} = 72.145773078307
x42=−40.6440368534098x_{42} = -40.6440368534098
x43=11.9023123123852x_{43} = 11.9023123123852
x44=15.1847140801157x_{44} = 15.1847140801157
x45=−15.1847140801157x_{45} = -15.1847140801157
x46=−109.882946199505x_{46} = -109.882946199505
x47=−94.1628332382898x_{47} = -94.1628332382898
x48=87.8735702250483x_{48} = 87.8735702250483
x49=68.9991276649867x_{49} = 68.9991276649867
x50=46.9536140378669x_{50} = 46.9536140378669
x51=−24.8110526666222x_{51} = -24.8110526666222
x52=−34.3247248305535x_{52} = -34.3247248305535
x53=50.1059069640945x_{53} = 50.1059069640945
x54=−72.145773078307x_{54} = -72.145773078307
x55=8.51016282615891x_{55} = 8.51016282615891
x56=59.5559841675756x_{56} = 59.5559841675756
x57=37.4859072816634x_{57} = 37.4859072816634
x58=−78.4378470253956x_{58} = -78.4378470253956
x59=31.1595522204503x_{59} = 31.1595522204503
x60=−59.5559841675756x_{60} = -59.5559841675756
x61=−37.4859072816634x_{61} = -37.4859072816634
x62=81.5833695706461x_{62} = 81.5833695706461
x63=116.170070456264x_{63} = 116.170070456264
x64=−100.451334931507x_{64} = -100.451334931507
x65=−8.51016282615891x_{65} = -8.51016282615891
x66=−53.2569316655778x_{66} = -53.2569316655778
x67=27.989021578396x_{67} = 27.989021578396
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=0x_{1} = 0

lim⁡x→0−(−sin⁡(x)−8cos⁡(x)x+20sin⁡(x)x2x4)=−∞\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} - \frac{8 \cos{\left(x \right)}}{x} + \frac{20 \sin{\left(x \right)}}{x^{2}}}{x^{4}}\right) = -\infty
lim⁡x→0+(−sin⁡(x)−8cos⁡(x)x+20sin⁡(x)x2x4)=∞\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} - \frac{8 \cos{\left(x \right)}}{x} + \frac{20 \sin{\left(x \right)}}{x^{2}}}{x^{4}}\right) = \infty
- the limits are not equal, so
x1=0x_{1} = 0
- is an inflection 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
[116.170070456264,∞)\left[116.170070456264, \infty\right)
Convex at the intervals
(−∞,−505.780600175478]\left(-\infty, -505.780600175478\right]
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡(x)x4)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x^{4}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(sin⁡(x)x4)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x^{4}}\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(x)/x^4, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)xx4)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x x^{4}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)xx4)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x x^{4}}\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(-x) и f = -f(-x).
So, check:
sin⁡(x)x4=−sin⁡(x)x4\frac{\sin{\left(x \right)}}{x^{4}} = - \frac{\sin{\left(x \right)}}{x^{4}}
- No
sin⁡(x)x4=sin⁡(x)x4\frac{\sin{\left(x \right)}}{x^{4}} = \frac{\sin{\left(x \right)}}{x^{4}}
- No
so, the function
not is
neither even, nor odd
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