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Plotting a function graph/ cos(x)/x

Graphing y = cos(x)/x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(x)
f(x) = ------
         x
f(x)=cos⁡(x)xf{\left(x \right)} = \frac{\cos{\left(x \right)}}{x} 
f = cos(x)/x
The graph of the function
02468-8-6-4-2-1010-20002000
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)x=0\frac{\cos{\left(x \right)}}{x} = 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=4.71238898038469x_{1} = 4.71238898038469
x2=17.2787595947439x_{2} = 17.2787595947439
x3=−89.5353906273091x_{3} = -89.5353906273091
x4=64.4026493985908x_{4} = 64.4026493985908
x5=70.6858347057703x_{5} = 70.6858347057703
x6=36.1283155162826x_{6} = 36.1283155162826
x7=−98.9601685880785x_{7} = -98.9601685880785
x8=48.6946861306418x_{8} = 48.6946861306418
x9=−58.1194640914112x_{9} = -58.1194640914112
x10=7.85398163397448x_{10} = 7.85398163397448
x11=39.2699081698724x_{11} = 39.2699081698724
x12=−422.544211907827x_{12} = -422.544211907827
x13=−95.8185759344887x_{13} = -95.8185759344887
x14=−1.5707963267949x_{14} = -1.5707963267949
x15=−92.6769832808989x_{15} = -92.6769832808989
x16=−23.5619449019235x_{16} = -23.5619449019235
x17=23.5619449019235x_{17} = 23.5619449019235
x18=61.261056745001x_{18} = 61.261056745001
x19=29.845130209103x_{19} = 29.845130209103
x20=−32.9867228626928x_{20} = -32.9867228626928
x21=−51.8362787842316x_{21} = -51.8362787842316
x22=−80.1106126665397x_{22} = -80.1106126665397
x23=−83.2522053201295x_{23} = -83.2522053201295
x24=67.5442420521806x_{24} = 67.5442420521806
x25=98.9601685880785x_{25} = 98.9601685880785
x26=92.6769832808989x_{26} = 92.6769832808989
x27=−39.2699081698724x_{27} = -39.2699081698724
x28=86.3937979737193x_{28} = 86.3937979737193
x29=45.553093477052x_{29} = 45.553093477052
x30=−67.5442420521806x_{30} = -67.5442420521806
x31=51.8362787842316x_{31} = 51.8362787842316
x32=76.9690200129499x_{32} = 76.9690200129499
x33=−26.7035375555132x_{33} = -26.7035375555132
x34=1173.38485611579x_{34} = 1173.38485611579
x35=347.145988221672x_{35} = 347.145988221672
x36=−4.71238898038469x_{36} = -4.71238898038469
x37=95.8185759344887x_{37} = 95.8185759344887
x38=−86.3937979737193x_{38} = -86.3937979737193
x39=−10.9955742875643x_{39} = -10.9955742875643
x40=83.2522053201295x_{40} = 83.2522053201295
x41=−7.85398163397448x_{41} = -7.85398163397448
x42=−36.1283155162826x_{42} = -36.1283155162826
x43=−17.2787595947439x_{43} = -17.2787595947439
x44=199.491133502952x_{44} = 199.491133502952
x45=20.4203522483337x_{45} = 20.4203522483337
x46=−14.1371669411541x_{46} = -14.1371669411541
x47=54.9778714378214x_{47} = 54.9778714378214
x48=−70.6858347057703x_{48} = -70.6858347057703
x49=−48.6946861306418x_{49} = -48.6946861306418
x50=−54.9778714378214x_{50} = -54.9778714378214
x51=−45.553093477052x_{51} = -45.553093477052
x52=14.1371669411541x_{52} = 14.1371669411541
x53=−73.8274273593601x_{53} = -73.8274273593601
x54=26.7035375555132x_{54} = 26.7035375555132
x55=89.5353906273091x_{55} = 89.5353906273091
x56=10.9955742875643x_{56} = 10.9955742875643
x57=80.1106126665397x_{57} = 80.1106126665397
x58=73.8274273593601x_{58} = 73.8274273593601
x59=58.1194640914112x_{59} = 58.1194640914112
x60=−61.261056745001x_{60} = -61.261056745001
x61=1.5707963267949x_{61} = 1.5707963267949
x62=−20.4203522483337x_{62} = -20.4203522483337
x63=−42.4115008234622x_{63} = -42.4115008234622
x64=32.9867228626928x_{64} = 32.9867228626928
x65=42.4115008234622x_{65} = 42.4115008234622
x66=−76.9690200129499x_{66} = -76.9690200129499
x67=−64.4026493985908x_{67} = -64.4026493985908
x68=−29.845130209103x_{68} = -29.845130209103
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.
cos⁡(0)0\frac{\cos{\left(0 \right)}}{0}
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)x−cos⁡(x)x2=0- \frac{\sin{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=47.1026627703624x_{1} = 47.1026627703624
x2=84.811211299318x_{2} = 84.811211299318
x3=−50.2455828375744x_{3} = -50.2455828375744
x4=−100.521017074687x_{4} = -100.521017074687
x5=−65.9582857893902x_{5} = -65.9582857893902
x6=65.9582857893902x_{6} = 65.9582857893902
x7=−78.5270825679419x_{7} = -78.5270825679419
x8=−21.945612879981x_{8} = -21.945612879981
x9=28.2389365752603x_{9} = 28.2389365752603
x10=53.3883466217256x_{10} = 53.3883466217256
x11=−87.9532251106725x_{11} = -87.9532251106725
x12=−25.0929104121121x_{12} = -25.0929104121121
x13=−34.5285657554621x_{13} = -34.5285657554621
x14=−69.100567727981x_{14} = -69.100567727981
x15=94.2371684817036x_{15} = 94.2371684817036
x16=−135.08108127842x_{16} = -135.08108127842
x17=−62.8159348889734x_{17} = -62.8159348889734
x18=50.2455828375744x_{18} = 50.2455828375744
x19=−91.0952098694071x_{19} = -91.0952098694071
x20=91.0952098694071x_{20} = 91.0952098694071
x21=56.5309801938186x_{21} = 56.5309801938186
x22=−53.3883466217256x_{22} = -53.3883466217256
x23=−169.640108529775x_{23} = -169.640108529775
x24=−109.946647805931x_{24} = -109.946647805931
x25=−6.12125046689807x_{25} = -6.12125046689807
x26=2.79838604578389x_{26} = 2.79838604578389
x27=−47.1026627703624x_{27} = -47.1026627703624
x28=62.8159348889734x_{28} = 62.8159348889734
x29=9.31786646179107x_{29} = 9.31786646179107
x30=−2.79838604578389x_{30} = -2.79838604578389
x31=−81.6691650818489x_{31} = -81.6691650818489
x32=12.4864543952238x_{32} = 12.4864543952238
x33=−31.3840740178899x_{33} = -31.3840740178899
x34=−94.2371684817036x_{34} = -94.2371684817036
x35=197.91528455229x_{35} = 197.91528455229
x36=59.6735041304405x_{36} = 59.6735041304405
x37=97.3791034786112x_{37} = 97.3791034786112
x38=75.3849592185347x_{38} = 75.3849592185347
x39=−40.8162093266346x_{39} = -40.8162093266346
x40=−15.644128370333x_{40} = -15.644128370333
x41=−37.672573565113x_{41} = -37.672573565113
x42=−12.4864543952238x_{42} = -12.4864543952238
x43=15.644128370333x_{43} = 15.644128370333
x44=69.100567727981x_{44} = 69.100567727981
x45=−84.811211299318x_{45} = -84.811211299318
x46=31.3840740178899x_{46} = 31.3840740178899
x47=37.672573565113x_{47} = 37.672573565113
x48=−97.3791034786112x_{48} = -97.3791034786112
x49=6.12125046689807x_{49} = 6.12125046689807
x50=72.2427897046973x_{50} = 72.2427897046973
x51=−75.3849592185347x_{51} = -75.3849592185347
x52=34.5285657554621x_{52} = 34.5285657554621
x53=43.9595528888955x_{53} = 43.9595528888955
x54=78.5270825679419x_{54} = 78.5270825679419
x55=40.8162093266346x_{55} = 40.8162093266346
x56=100.521017074687x_{56} = 100.521017074687
x57=21.945612879981x_{57} = 21.945612879981
x58=−9.31786646179107x_{58} = -9.31786646179107
x59=25.0929104121121x_{59} = 25.0929104121121
x60=−72.2427897046973x_{60} = -72.2427897046973
x61=−18.7964043662102x_{61} = -18.7964043662102
x62=87.9532251106725x_{62} = 87.9532251106725
x63=−59.6735041304405x_{63} = -59.6735041304405
x64=−28.2389365752603x_{64} = -28.2389365752603
x65=18.7964043662102x_{65} = 18.7964043662102
x66=81.6691650818489x_{66} = 81.6691650818489
x67=−56.5309801938186x_{67} = -56.5309801938186
x68=−43.9595528888955x_{68} = -43.9595528888955
The values of the extrema at the points:
(47.10266277036235, -0.0212254394164143)

(84.81121129931802, -0.0117900744410766)

(-50.24558283757444, -0.0198983065303553)

(-100.52101707468658, -0.00994767611536293)

(-65.95828578939016, 0.0151593553168405)

(65.95828578939016, -0.0151593553168405)

(-78.52708256794193, 0.0127334276777468)

(-21.945612879981045, 0.0455199604051285)

(28.238936575260272, -0.0353899155541688)

(53.38834662172563, -0.0187273944640866)

(-87.95322511067255, -0.0113689449158811)

(-25.092910412112097, -0.0398202855500511)

(-34.52856575546206, 0.0289493889114503)

(-69.10056772798097, -0.0144701459746764)

(94.23716848170359, 0.01061092686295)

(-135.0810812784199, 0.00740275832666827)

(-62.81593488897342, -0.015917510583426)

(50.24558283757444, 0.0198983065303553)

(-91.09520986940714, 0.0109768642483425)

(91.09520986940714, -0.0109768642483425)

(56.53098019381864, 0.0176866485521696)

(-53.38834662172563, 0.0187273944640866)

(-169.6401085297751, -0.00589472993500857)

(-109.94664780593057, 0.00909494432157336)

(-6.1212504668980685, -0.161228034325064)

(2.798386045783887, -0.336508416918395)

(-47.10266277036235, 0.0212254394164143)

(62.81593488897342, 0.015917510583426)

(9.317866461791066, -0.106707947715237)

(-2.798386045783887, 0.336508416918395)

(-81.66916508184887, -0.0122436055670467)

(12.486454395223781, 0.0798311807800032)

(-31.38407401788986, -0.0318471321112693)

(-94.23716848170359, -0.01061092686295)

(197.91528455229027, -0.00505260236866135)

(59.67350413044053, -0.0167555036571887)

(97.3791034786112, -0.0102686022030809)

(75.38495921853475, 0.0132640786518247)

(-40.81620932663458, 0.0244927205346957)

(-15.644128370333028, 0.0637915530395936)

(-37.67257356511297, -0.0265351630103045)

(-12.486454395223781, -0.0798311807800032)

(15.644128370333028, -0.0637915530395936)

(69.10056772798097, 0.0144701459746764)

(-84.81121129931802, 0.0117900744410766)

(31.38407401788986, 0.0318471321112693)

(37.67257356511297, 0.0265351630103045)

(-97.3791034786112, 0.0102686022030809)

(6.1212504668980685, 0.161228034325064)

(72.24278970469729, -0.0138408859131547)

(-75.38495921853475, -0.0132640786518247)

(34.52856575546206, -0.0289493889114503)

(43.959552888895495, 0.0227423004725314)

(78.52708256794193, -0.0127334276777468)

(40.81620932663458, -0.0244927205346957)

(100.52101707468658, 0.00994767611536293)

(21.945612879981045, -0.0455199604051285)

(-9.317866461791066, 0.106707947715237)

(25.092910412112097, 0.0398202855500511)

(-72.24278970469729, 0.0138408859131547)

(-18.796404366210158, -0.0531265325613881)

(87.95322511067255, 0.0113689449158811)

(-59.67350413044053, 0.0167555036571887)

(-28.238936575260272, 0.0353899155541688)

(18.796404366210158, 0.0531265325613881)

(81.66916508184887, 0.0122436055670467)

(-56.53098019381864, -0.0176866485521696)

(-43.959552888895495, -0.0227423004725314)


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=47.1026627703624x_{1} = 47.1026627703624
x2=84.811211299318x_{2} = 84.811211299318
x3=−50.2455828375744x_{3} = -50.2455828375744
x4=−100.521017074687x_{4} = -100.521017074687
x5=65.9582857893902x_{5} = 65.9582857893902
x6=28.2389365752603x_{6} = 28.2389365752603
x7=53.3883466217256x_{7} = 53.3883466217256
x8=−87.9532251106725x_{8} = -87.9532251106725
x9=−25.0929104121121x_{9} = -25.0929104121121
x10=−69.100567727981x_{10} = -69.100567727981
x11=−62.8159348889734x_{11} = -62.8159348889734
x12=91.0952098694071x_{12} = 91.0952098694071
x13=−169.640108529775x_{13} = -169.640108529775
x14=−6.12125046689807x_{14} = -6.12125046689807
x15=2.79838604578389x_{15} = 2.79838604578389
x16=9.31786646179107x_{16} = 9.31786646179107
x17=−81.6691650818489x_{17} = -81.6691650818489
x18=−31.3840740178899x_{18} = -31.3840740178899
x19=−94.2371684817036x_{19} = -94.2371684817036
x20=197.91528455229x_{20} = 197.91528455229
x21=59.6735041304405x_{21} = 59.6735041304405
x22=97.3791034786112x_{22} = 97.3791034786112
x23=−37.672573565113x_{23} = -37.672573565113
x24=−12.4864543952238x_{24} = -12.4864543952238
x25=15.644128370333x_{25} = 15.644128370333
x26=72.2427897046973x_{26} = 72.2427897046973
x27=−75.3849592185347x_{27} = -75.3849592185347
x28=34.5285657554621x_{28} = 34.5285657554621
x29=78.5270825679419x_{29} = 78.5270825679419
x30=40.8162093266346x_{30} = 40.8162093266346
x31=21.945612879981x_{31} = 21.945612879981
x32=−18.7964043662102x_{32} = -18.7964043662102
x33=−56.5309801938186x_{33} = -56.5309801938186
x34=−43.9595528888955x_{34} = -43.9595528888955
Maxima of the function at points:
x34=−65.9582857893902x_{34} = -65.9582857893902
x34=−78.5270825679419x_{34} = -78.5270825679419
x34=−21.945612879981x_{34} = -21.945612879981
x34=−34.5285657554621x_{34} = -34.5285657554621
x34=94.2371684817036x_{34} = 94.2371684817036
x34=−135.08108127842x_{34} = -135.08108127842
x34=50.2455828375744x_{34} = 50.2455828375744
x34=−91.0952098694071x_{34} = -91.0952098694071
x34=56.5309801938186x_{34} = 56.5309801938186
x34=−53.3883466217256x_{34} = -53.3883466217256
x34=−109.946647805931x_{34} = -109.946647805931
x34=−47.1026627703624x_{34} = -47.1026627703624
x34=62.8159348889734x_{34} = 62.8159348889734
x34=−2.79838604578389x_{34} = -2.79838604578389
x34=12.4864543952238x_{34} = 12.4864543952238
x34=75.3849592185347x_{34} = 75.3849592185347
x34=−40.8162093266346x_{34} = -40.8162093266346
x34=−15.644128370333x_{34} = -15.644128370333
x34=69.100567727981x_{34} = 69.100567727981
x34=−84.811211299318x_{34} = -84.811211299318
x34=31.3840740178899x_{34} = 31.3840740178899
x34=37.672573565113x_{34} = 37.672573565113
x34=−97.3791034786112x_{34} = -97.3791034786112
x34=6.12125046689807x_{34} = 6.12125046689807
x34=43.9595528888955x_{34} = 43.9595528888955
x34=100.521017074687x_{34} = 100.521017074687
x34=−9.31786646179107x_{34} = -9.31786646179107
x34=25.0929104121121x_{34} = 25.0929104121121
x34=−72.2427897046973x_{34} = -72.2427897046973
x34=87.9532251106725x_{34} = 87.9532251106725
x34=−59.6735041304405x_{34} = -59.6735041304405
x34=−28.2389365752603x_{34} = -28.2389365752603
x34=18.7964043662102x_{34} = 18.7964043662102
x34=81.6691650818489x_{34} = 81.6691650818489
Decreasing at intervals
[197.91528455229,∞)\left[197.91528455229, \infty\right)
Increasing at intervals
(−∞,−169.640108529775]\left(-\infty, -169.640108529775\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)+2sin⁡(x)x+2cos⁡(x)x2x=0\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \right)}}{x^{2}}}{x} = 0
Solve this equation
The roots of this equation
x1=64.3715747870554x_{1} = 64.3715747870554
x2=−92.655396245836x_{2} = -92.655396245836
x3=−70.6575253785884x_{3} = -70.6575253785884
x4=95.7976970894915x_{4} = 95.7976970894915
x5=−13.9937625671267x_{5} = -13.9937625671267
x6=80.0856368040887x_{6} = 80.0856368040887
x7=61.2283863503723x_{7} = 61.2283863503723
x8=−32.9259431758392x_{8} = -32.9259431758392
x9=20.3217772482235x_{9} = 20.3217772482235
x10=−54.9414610202918x_{10} = -54.9414610202918
x11=29.7779159141436x_{11} = 29.7779159141436
x12=−76.9430238267933x_{12} = -76.9430238267933
x13=45.5091321154553x_{13} = 45.5091321154553
x14=98.9399529307048x_{14} = 98.9399529307048
x15=−83.2281726832512x_{15} = -83.2281726832512
x16=−95.7976970894915x_{16} = -95.7976970894915
x17=230.898398112111x_{17} = 230.898398112111
x18=48.6535676048409x_{18} = 48.6535676048409
x19=89.5130456566371x_{19} = 89.5130456566371
x20=7.5873993379941x_{20} = 7.5873993379941
x21=36.0728437679879x_{21} = 36.0728437679879
x22=−20.3217772482235x_{22} = -20.3217772482235
x23=42.3642737086586x_{23} = 42.3642737086586
x24=−23.4766510546492x_{24} = -23.4766510546492
x25=−39.218890250481x_{25} = -39.218890250481
x26=−29.7779159141436x_{26} = -29.7779159141436
x27=73.8003238908837x_{27} = 73.8003238908837
x28=10.8095072981602x_{28} = 10.8095072981602
x29=67.5146145048817x_{29} = 67.5146145048817
x30=−73.8003238908837x_{30} = -73.8003238908837
x31=−10.8095072981602x_{31} = -10.8095072981602
x32=58.085025007445x_{32} = 58.085025007445
x33=92.655396245836x_{33} = 92.655396245836
x34=−86.370639887736x_{34} = -86.370639887736
x35=−271.740404503579x_{35} = -271.740404503579
x36=86.370639887736x_{36} = 86.370639887736
x37=−58.085025007445x_{37} = -58.085025007445
x38=32.9259431758392x_{38} = 32.9259431758392
x39=−67.5146145048817x_{39} = -67.5146145048817
x40=−89.5130456566371x_{40} = -89.5130456566371
x41=−45.5091321154553x_{41} = -45.5091321154553
x42=4.2222763997912x_{42} = 4.2222763997912
x43=−51.7976574095537x_{43} = -51.7976574095537
x44=23.4766510546492x_{44} = 23.4766510546492
x45=−64.3715747870554x_{45} = -64.3715747870554
x46=26.6283591640252x_{46} = 26.6283591640252
x47=39.218890250481x_{47} = 39.218890250481
x48=−80.0856368040887x_{48} = -80.0856368040887
x49=54.9414610202918x_{49} = 54.9414610202918
x50=−26.6283591640252x_{50} = -26.6283591640252
x51=−7.5873993379941x_{51} = -7.5873993379941
x52=76.9430238267933x_{52} = 76.9430238267933
x53=−36.0728437679879x_{53} = -36.0728437679879
x54=−48.6535676048409x_{54} = -48.6535676048409
x55=51.7976574095537x_{55} = 51.7976574095537
x56=−98.9399529307048x_{56} = -98.9399529307048
x57=70.6575253785884x_{57} = 70.6575253785884
x58=17.1619600917303x_{58} = 17.1619600917303
x59=83.2281726832512x_{59} = 83.2281726832512
x60=−61.2283863503723x_{60} = -61.2283863503723
x61=−17.1619600917303x_{61} = -17.1619600917303
x62=−42.3642737086586x_{62} = -42.3642737086586
x63=13.9937625671267x_{63} = 13.9937625671267
x64=−4.2222763997912x_{64} = -4.2222763997912
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)+2sin⁡(x)x+2cos⁡(x)x2x)=−∞\lim_{x \to 0^-}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \right)}}{x^{2}}}{x}\right) = -\infty
lim⁡x→0+(−cos⁡(x)+2sin⁡(x)x+2cos⁡(x)x2x)=∞\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \right)}}{x^{2}}}{x}\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
[95.7976970894915,∞)\left[95.7976970894915, \infty\right)
Convex at the intervals
(−∞,−271.740404503579]\left(-\infty, -271.740404503579\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)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x}\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, divided by x 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,
inclined coincides with the horizontal asymptote on the right
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,
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)x=−cos⁡(x)x\frac{\cos{\left(x \right)}}{x} = - \frac{\cos{\left(x \right)}}{x}
- No
cos⁡(x)x=cos⁡(x)x\frac{\cos{\left(x \right)}}{x} = \frac{\cos{\left(x \right)}}{x}
- No
so, the function
not is
neither even, nor odd
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