Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • x√2-x
  • x^3-12x+24
  • x^2+6x+5
  • -x^2+5*x+4
  • Integral of d{x}:
  • cos(x)/x^2 cos(x)/x^2
  • Derivative of:
  • cos(x)/x^2 cos(x)/x^2
  • Limit of the function:
  • cos(x)/x^2 cos(x)/x^2

  • Identical expressions

  • cos(x)/x^ two
  • co sinus of e of (x) divide by x squared
  • co sinus of e of (x) divide by x to the power of two
  • cos(x)/x2
  • cosx/x2
  • cos(x)/x²
  • cos(x)/x to the power of 2
  • cosx/x^2
  • cos(x) divide by x^2

  • Similar expressions

  • cosx/x^2
Plotting a function graph/ cos(x)/x^2

Graphing y = cos(x)/x^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(x)
f(x) = ------
          2  
         x
f(x)=cos⁡(x)x2f{\left(x \right)} = \frac{\cos{\left(x \right)}}{x^{2}} 
f = cos(x)/x^2
The graph of the function
-0.50-0.40-0.30-0.20-0.100.500.000.100.200.300.4002000000
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:
cos⁡(x)x2=0\frac{\cos{\left(x \right)}}{x^{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=17.2787595947439x_{1} = 17.2787595947439
x2=−23.5619449019235x_{2} = -23.5619449019235
x3=−14.1371669411541x_{3} = -14.1371669411541
x4=51.8362787842316x_{4} = 51.8362787842316
x5=−17.2787595947439x_{5} = -17.2787595947439
x6=−10.9955742875643x_{6} = -10.9955742875643
x7=−36.1283155162826x_{7} = -36.1283155162826
x8=−95.8185759344887x_{8} = -95.8185759344887
x9=−48.6946861306418x_{9} = -48.6946861306418
x10=−26.7035375555132x_{10} = -26.7035375555132
x11=4.71238898038469x_{11} = 4.71238898038469
x12=26.7035375555132x_{12} = 26.7035375555132
x13=89.5353906273091x_{13} = 89.5353906273091
x14=23.5619449019235x_{14} = 23.5619449019235
x15=14.1371669411541x_{15} = 14.1371669411541
x16=42.4115008234622x_{16} = 42.4115008234622
x17=95.8185759344887x_{17} = 95.8185759344887
x18=−61.261056745001x_{18} = -61.261056745001
x19=58.1194640914112x_{19} = 58.1194640914112
x20=186.924762888593x_{20} = 186.924762888593
x21=36.1283155162826x_{21} = 36.1283155162826
x22=29.845130209103x_{22} = 29.845130209103
x23=−73.8274273593601x_{23} = -73.8274273593601
x24=61.261056745001x_{24} = 61.261056745001
x25=48.6946861306418x_{25} = 48.6946861306418
x26=−4.71238898038469x_{26} = -4.71238898038469
x27=70.6858347057703x_{27} = 70.6858347057703
x28=−7.85398163397448x_{28} = -7.85398163397448
x29=−51.8362787842316x_{29} = -51.8362787842316
x30=−76.9690200129499x_{30} = -76.9690200129499
x31=−89.5353906273091x_{31} = -89.5353906273091
x32=−39.2699081698724x_{32} = -39.2699081698724
x33=10.9955742875643x_{33} = 10.9955742875643
x34=−42.4115008234622x_{34} = -42.4115008234622
x35=80.1106126665397x_{35} = 80.1106126665397
x36=83.2522053201295x_{36} = 83.2522053201295
x37=−92.6769832808989x_{37} = -92.6769832808989
x38=32.9867228626928x_{38} = 32.9867228626928
x39=45.553093477052x_{39} = 45.553093477052
x40=20.4203522483337x_{40} = 20.4203522483337
x41=64.4026493985908x_{41} = 64.4026493985908
x42=−32.9867228626928x_{42} = -32.9867228626928
x43=67.5442420521806x_{43} = 67.5442420521806
x44=−20.4203522483337x_{44} = -20.4203522483337
x45=−80.1106126665397x_{45} = -80.1106126665397
x46=7.85398163397448x_{46} = 7.85398163397448
x47=−45.553093477052x_{47} = -45.553093477052
x48=76.9690200129499x_{48} = 76.9690200129499
x49=−1.5707963267949x_{49} = -1.5707963267949
x50=39.2699081698724x_{50} = 39.2699081698724
x51=−70.6858347057703x_{51} = -70.6858347057703
x52=−67.5442420521806x_{52} = -67.5442420521806
x53=−98.9601685880785x_{53} = -98.9601685880785
x54=−29.845130209103x_{54} = -29.845130209103
x55=−83.2522053201295x_{55} = -83.2522053201295
x56=−86.3937979737193x_{56} = -86.3937979737193
x57=98.9601685880785x_{57} = 98.9601685880785
x58=73.8274273593601x_{58} = 73.8274273593601
x59=−58.1194640914112x_{59} = -58.1194640914112
x60=92.6769832808989x_{60} = 92.6769832808989
x61=54.9778714378214x_{61} = 54.9778714378214
x62=86.3937979737193x_{62} = 86.3937979737193
x63=1.5707963267949x_{63} = 1.5707963267949
x64=−54.9778714378214x_{64} = -54.9778714378214
x65=−64.4026493985908x_{65} = -64.4026493985908
x66=108.384946548848x_{66} = 108.384946548848
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)/x^2.
cos⁡(0)02\frac{\cos{\left(0 \right)}}{0^{2}}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
−sin⁡(x)x2−2cos⁡(x)x3=0- \frac{\sin{\left(x \right)}}{x^{2}} - \frac{2 \cos{\left(x \right)}}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=34.4996123350132x_{1} = 34.4996123350132
x2=−5.95939190757933x_{2} = -5.95939190757933
x3=9.21096438740149x_{3} = 9.21096438740149
x4=−31.3522215217643x_{4} = -31.3522215217643
x5=56.5132926241755x_{5} = 56.5132926241755
x6=−56.5132926241755x_{6} = -56.5132926241755
x7=−59.6567478435559x_{7} = -59.6567478435559
x8=94.2265573558031x_{8} = 94.2265573558031
x9=−103.653264863525x_{9} = -103.653264863525
x10=−50.2256832197934x_{10} = -50.2256832197934
x11=−47.0814357397523x_{11} = -47.0814357397523
x12=91.0842327848165x_{12} = 91.0842327848165
x13=−25.053079662454x_{13} = -25.053079662454
x14=−53.3696181339615x_{14} = -53.3696181339615
x15=−37.6460352959305x_{15} = -37.6460352959305
x16=65.9431258539286x_{16} = 65.9431258539286
x17=−18.7432530945386x_{17} = -18.7432530945386
x18=53.3696181339615x_{18} = 53.3696181339615
x19=−12.4065403639626x_{19} = -12.4065403639626
x20=84.7994209518635x_{20} = 84.7994209518635
x21=81.6569211705466x_{21} = 81.6569211705466
x22=47.0814357397523x_{22} = 47.0814357397523
x23=−28.2035393053095x_{23} = -28.2035393053095
x24=37.6460352959305x_{24} = 37.6460352959305
x25=−15.5802941824244x_{25} = -15.5802941824244
x26=−91.0842327848165x_{26} = -91.0842327848165
x27=−81.6569211705466x_{27} = -81.6569211705466
x28=75.3716947511882x_{28} = 75.3716947511882
x29=25.053079662454x_{29} = 25.053079662454
x30=−78.5143487963623x_{30} = -78.5143487963623
x31=40.7917141624847x_{31} = 40.7917141624847
x32=−97.3688346960149x_{32} = -97.3688346960149
x33=−9.21096438740149x_{33} = -9.21096438740149
x34=−72.2289483771681x_{34} = -72.2289483771681
x35=135.073678452493x_{35} = 135.073678452493
x36=78.5143487963623x_{36} = 78.5143487963623
x37=−62.8000167068325x_{37} = -62.8000167068325
x38=−109.93755273626x_{38} = -109.93755273626
x39=28.2035393053095x_{39} = 28.2035393053095
x40=50.2256832197934x_{40} = 50.2256832197934
x41=−34.4996123350132x_{41} = -34.4996123350132
x42=72.2289483771681x_{42} = 72.2289483771681
x43=18.7432530945386x_{43} = 18.7432530945386
x44=12.4065403639626x_{44} = 12.4065403639626
x45=43.9368086315937x_{45} = 43.9368086315937
x46=−87.9418559209576x_{46} = -87.9418559209576
x47=100.511069234565x_{47} = 100.511069234565
x48=87.9418559209576x_{48} = 87.9418559209576
x49=21.9000773156394x_{49} = 21.9000773156394
x50=−21.9000773156394x_{50} = -21.9000773156394
x51=−94.2265573558031x_{51} = -94.2265573558031
x52=15.5802941824244x_{52} = 15.5802941824244
x53=59.6567478435559x_{53} = 59.6567478435559
x54=−100.511069234565x_{54} = -100.511069234565
x55=−75.3716947511882x_{55} = -75.3716947511882
x56=31.3522215217643x_{56} = 31.3522215217643
x57=−65.9431258539286x_{57} = -65.9431258539286
x58=−69.0860970774096x_{58} = -69.0860970774096
x59=69.0860970774096x_{59} = 69.0860970774096
x60=−43.9368086315937x_{60} = -43.9368086315937
x61=97.3688346960149x_{61} = 97.3688346960149
x62=−40.7917141624847x_{62} = -40.7917141624847
x63=5.95939190757933x_{63} = 5.95939190757933
x64=62.8000167068325x_{64} = 62.8000167068325
x65=−84.7994209518635x_{65} = -84.7994209518635
x66=2.45871417599962x_{66} = 2.45871417599962
x67=−2.45871417599962x_{67} = -2.45871417599962
The values of the extrema at the points:
(34.4996123350132, -0.000838770260526343)

(-5.9593919075793265, 0.0266944281300046)

(9.210964387401486, -0.0115182384102548)

(-31.352221521764292, 0.0010152698990766)

(56.513292624175506, 0.000312915432650295)

(-56.513292624175506, 0.000312915432650295)

(-59.656747843555884, -0.000280825751144458)

(94.22655735580307, 0.000112604447700661)

(-103.65326486352524, -9.30578895300905e-5)

(-50.2256832197934, 0.000396099456126142)

(-47.081435739752315, -0.000450722368032648)

(91.08423278481655, -0.000120506069272649)

(-25.053079662453992, 0.00158817477024791)

(-53.36961813396146, -0.000350838362181669)

(-37.64603529593052, 0.000704611119614408)

(65.94312585392862, -0.000229858880631)

(-18.74325309453857, 0.00283042465312132)

(53.36961813396146, -0.000350838362181669)

(-12.406540363962565, 0.00641398077993427)

(84.79942095186354, -0.000139025181535869)

(81.65692117054658, 0.000149928353545869)

(47.081435739752315, -0.000450722368032648)

(-28.20353930530947, -0.00125401736797822)

(37.64603529593052, 0.000704611119614408)

(-15.580294182424433, -0.00408601227579287)

(-91.08423278481655, -0.000120506069272649)

(-81.65692117054658, 0.000149928353545869)

(75.37169475118824, 0.000175966743144092)

(25.053079662453992, 0.00158817477024791)

(-78.51434879636227, -0.000162166475547147)

(40.79171416248471, -0.00060025351930421)

(-97.36883469601494, -0.000105455311245964)

(-9.210964387401486, -0.0115182384102548)

(-72.22894837716808, -0.00019160683134921)

(135.07367845249348, -5.48038341935653e-5)

(78.51434879636227, -0.000162166475547147)

(-62.80001670683253, 0.000253431359776371)

(-109.93755273625987, -8.27248552367837e-5)

(28.20353930530947, -0.00125401736797822)

(50.2256832197934, 0.000396099456126142)

(-34.4996123350132, -0.000838770260526343)

(72.22894837716808, -0.00019160683134921)

(18.74325309453857, 0.00283042465312132)

(12.406540363962565, 0.00641398077993427)

(43.936808631593706, 0.000517479923876906)

(-87.94185592095755, 0.000129269617694298)

(100.51106923456473, 9.89660537297585e-5)

(87.94185592095755, 0.000129269617694298)

(21.90007731563936, -0.00207637214990232)

(-21.90007731563936, -0.00207637214990232)

(-94.22655735580307, 0.000112604447700661)

(15.580294182424433, -0.00408601227579287)

(59.656747843555884, -0.000280825751144458)

(-100.51106923456473, 9.89660537297585e-5)

(-75.37169475118824, 0.000175966743144092)

(31.352221521764292, 0.0010152698990766)

(-65.94312585392862, -0.000229858880631)

(-69.08609707740959, 0.000209428978902002)

(69.08609707740959, 0.000209428978902002)

(-43.936808631593706, 0.000517479923876906)

(97.36883469601494, -0.000105455311245964)

(-40.79171416248471, -0.00060025351930421)

(5.9593919075793265, 0.0266944281300046)

(62.80001670683253, 0.000253431359776371)

(-84.79942095186354, -0.000139025181535869)

(2.4587141759996247, -0.128324928485094)

(-2.4587141759996247, -0.128324928485094)


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=34.4996123350132x_{1} = 34.4996123350132
x2=9.21096438740149x_{2} = 9.21096438740149
x3=−59.6567478435559x_{3} = -59.6567478435559
x4=−103.653264863525x_{4} = -103.653264863525
x5=−47.0814357397523x_{5} = -47.0814357397523
x6=91.0842327848165x_{6} = 91.0842327848165
x7=−53.3696181339615x_{7} = -53.3696181339615
x8=65.9431258539286x_{8} = 65.9431258539286
x9=53.3696181339615x_{9} = 53.3696181339615
x10=84.7994209518635x_{10} = 84.7994209518635
x11=47.0814357397523x_{11} = 47.0814357397523
x12=−28.2035393053095x_{12} = -28.2035393053095
x13=−15.5802941824244x_{13} = -15.5802941824244
x14=−91.0842327848165x_{14} = -91.0842327848165
x15=−78.5143487963623x_{15} = -78.5143487963623
x16=40.7917141624847x_{16} = 40.7917141624847
x17=−97.3688346960149x_{17} = -97.3688346960149
x18=−9.21096438740149x_{18} = -9.21096438740149
x19=−72.2289483771681x_{19} = -72.2289483771681
x20=135.073678452493x_{20} = 135.073678452493
x21=78.5143487963623x_{21} = 78.5143487963623
x22=−109.93755273626x_{22} = -109.93755273626
x23=28.2035393053095x_{23} = 28.2035393053095
x24=−34.4996123350132x_{24} = -34.4996123350132
x25=72.2289483771681x_{25} = 72.2289483771681
x26=21.9000773156394x_{26} = 21.9000773156394
x27=−21.9000773156394x_{27} = -21.9000773156394
x28=15.5802941824244x_{28} = 15.5802941824244
x29=59.6567478435559x_{29} = 59.6567478435559
x30=−65.9431258539286x_{30} = -65.9431258539286
x31=97.3688346960149x_{31} = 97.3688346960149
x32=−40.7917141624847x_{32} = -40.7917141624847
x33=−84.7994209518635x_{33} = -84.7994209518635
x34=2.45871417599962x_{34} = 2.45871417599962
x35=−2.45871417599962x_{35} = -2.45871417599962
Maxima of the function at points:
x35=−5.95939190757933x_{35} = -5.95939190757933
x35=−31.3522215217643x_{35} = -31.3522215217643
x35=56.5132926241755x_{35} = 56.5132926241755
x35=−56.5132926241755x_{35} = -56.5132926241755
x35=94.2265573558031x_{35} = 94.2265573558031
x35=−50.2256832197934x_{35} = -50.2256832197934
x35=−25.053079662454x_{35} = -25.053079662454
x35=−37.6460352959305x_{35} = -37.6460352959305
x35=−18.7432530945386x_{35} = -18.7432530945386
x35=−12.4065403639626x_{35} = -12.4065403639626
x35=81.6569211705466x_{35} = 81.6569211705466
x35=37.6460352959305x_{35} = 37.6460352959305
x35=−81.6569211705466x_{35} = -81.6569211705466
x35=75.3716947511882x_{35} = 75.3716947511882
x35=25.053079662454x_{35} = 25.053079662454
x35=−62.8000167068325x_{35} = -62.8000167068325
x35=50.2256832197934x_{35} = 50.2256832197934
x35=18.7432530945386x_{35} = 18.7432530945386
x35=12.4065403639626x_{35} = 12.4065403639626
x35=43.9368086315937x_{35} = 43.9368086315937
x35=−87.9418559209576x_{35} = -87.9418559209576
x35=100.511069234565x_{35} = 100.511069234565
x35=87.9418559209576x_{35} = 87.9418559209576
x35=−94.2265573558031x_{35} = -94.2265573558031
x35=−100.511069234565x_{35} = -100.511069234565
x35=−75.3716947511882x_{35} = -75.3716947511882
x35=31.3522215217643x_{35} = 31.3522215217643
x35=−69.0860970774096x_{35} = -69.0860970774096
x35=69.0860970774096x_{35} = 69.0860970774096
x35=−43.9368086315937x_{35} = -43.9368086315937
x35=5.95939190757933x_{35} = 5.95939190757933
x35=62.8000167068325x_{35} = 62.8000167068325
Decreasing at intervals
[135.073678452493,∞)\left[135.073678452493, \infty\right)
Increasing at intervals
(−∞,−109.93755273626]\left(-\infty, -109.93755273626\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
−cos⁡(x)+4sin⁡(x)x+6cos⁡(x)x2x2=0\frac{- \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{x} + \frac{6 \cos{\left(x \right)}}{x^{2}}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=−29.7103970410056x_{1} = -29.7103970410056
x2=61.1956810599968x_{2} = 61.1956810599968
x3=318.859109531625x_{3} = 318.859109531625
x4=89.4906895254562x_{4} = 89.4906895254562
x5=95.7768091402129x_{5} = 95.7768091402129
x6=−17.0435524410056x_{6} = -17.0435524410056
x7=−73.7732005024933x_{7} = -73.7732005024933
x8=397.401405246951x_{8} = 397.401405246951
x9=67.4849609355352x_{9} = 67.4849609355352
x10=20.2222381068518x_{10} = 20.2222381068518
x11=54.9050022900618x_{11} = 54.9050022900618
x12=−92.6337991480858x_{12} = -92.6337991480858
x13=−36.0172009751693x_{13} = -36.0172009751693
x14=−64.3404701495554x_{14} = -64.3404701495554
x15=42.3169411013932x_{15} = 42.3169411013932
x16=−54.9050022900618x_{16} = -54.9050022900618
x17=−20.2222381068518x_{17} = -20.2222381068518
x18=−13.8473675657494x_{18} = -13.8473675657494
x19=29.7103970410056x_{19} = 29.7103970410056
x20=−51.7589783694261x_{20} = -51.7589783694261
x21=−7.3008858832282x_{21} = -7.3008858832282
x22=−83.2041261605509x_{22} = -83.2041261605509
x23=−39.167739309003x_{23} = -39.167739309003
x24=98.9197290094896x_{24} = 98.9197290094896
x25=32.8649384028032x_{25} = 32.8649384028032
x26=17.0435524410056x_{26} = 17.0435524410056
x27=−48.6123794800497x_{27} = -48.6123794800497
x28=10.6168428630359x_{28} = 10.6168428630359
x29=−86.3474693776984x_{29} = -86.3474693776984
x30=−42.3169411013932x_{30} = -42.3169411013932
x31=73.7732005024933x_{31} = 73.7732005024933
x32=26.5527542671168x_{32} = 26.5527542671168
x33=105.205330720799x_{33} = 105.205330720799
x34=−58.0505450446381x_{34} = -58.0505450446381
x35=−23.3907336526907x_{35} = -23.3907336526907
x36=58.0505450446381x_{36} = 58.0505450446381
x37=−89.4906895254562x_{37} = -89.4906895254562
x38=92.6337991480858x_{38} = 92.6337991480858
x39=83.2041261605509x_{39} = 83.2041261605509
x40=−70.6291933516676x_{40} = -70.6291933516676
x41=36.0172009751693x_{41} = 36.0172009751693
x42=−10.6168428630359x_{42} = -10.6168428630359
x43=−32.8649384028032x_{43} = -32.8649384028032
x44=−3.58835703497019x_{44} = -3.58835703497019
x45=−45.4650856843323x_{45} = -45.4650856843323
x46=−98.9197290094896x_{46} = -98.9197290094896
x47=3.58835703497019x_{47} = 3.58835703497019
x48=−26.5527542671168x_{48} = -26.5527542671168
x49=−95.7768091402129x_{49} = -95.7768091402129
x50=7.3008858832282x_{50} = 7.3008858832282
x51=86.3474693776984x_{51} = 86.3474693776984
x52=76.9170100644479x_{52} = 76.9170100644479
x53=48.6123794800497x_{53} = 48.6123794800497
x54=70.6291933516676x_{54} = 70.6291933516676
x55=39.167739309003x_{55} = 39.167739309003
x56=−67.4849609355352x_{56} = -67.4849609355352
x57=−80.0606453553676x_{57} = -80.0606453553676
x58=64.3404701495554x_{58} = 64.3404701495554
x59=13.8473675657494x_{59} = 13.8473675657494
x60=80.0606453553676x_{60} = 80.0606453553676
x61=51.7589783694261x_{61} = 51.7589783694261
x62=−124.060666131211x_{62} = -124.060666131211
x63=−61.1956810599968x_{63} = -61.1956810599968
x64=45.4650856843323x_{64} = 45.4650856843323
x65=23.3907336526907x_{65} = 23.3907336526907
x66=−76.9170100644479x_{66} = -76.9170100644479
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−(−cos⁡(x)+4sin⁡(x)x+6cos⁡(x)x2x2)=∞\lim_{x \to 0^-}\left(\frac{- \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{x} + \frac{6 \cos{\left(x \right)}}{x^{2}}}{x^{2}}\right) = \infty
lim⁡x→0+(−cos⁡(x)+4sin⁡(x)x+6cos⁡(x)x2x2)=∞\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{x} + \frac{6 \cos{\left(x \right)}}{x^{2}}}{x^{2}}\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
[397.401405246951,∞)\left[397.401405246951, \infty\right)
Convex at the intervals
(−∞,−124.060666131211]\left(-\infty, -124.060666131211\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→−∞(cos⁡(x)x2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(cos⁡(x)x2)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x^{2}}\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 cos(x)/x^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x)xx2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x)xx2)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x x^{2}}\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:
cos⁡(x)x2=cos⁡(x)x2\frac{\cos{\left(x \right)}}{x^{2}} = \frac{\cos{\left(x \right)}}{x^{2}}
- Yes
cos⁡(x)x2=−cos⁡(x)x2\frac{\cos{\left(x \right)}}{x^{2}} = - \frac{\cos{\left(x \right)}}{x^{2}}
- No
so, the function
is
even
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