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Plotting a function graph/ cos(n)/n

Graphing y = cos(n)/n

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(n)
f(n) = ------
         n
f(n)=cos⁡(n)nf{\left(n \right)} = \frac{\cos{\left(n \right)}}{n} 
f = cos(n)/n
The graph of the function
02468-8-6-4-2-1010-5050
The domain of the function
The points at which the function is not precisely defined:
n1=0n_{1} = 0
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis N at f = 0
so we need to solve the equation:
cos⁡(n)n=0\frac{\cos{\left(n \right)}}{n} = 0
Solve this equation
The points of intersection with the axis N:

Analytical solution
n1=π2n_{1} = \frac{\pi}{2}
n2=3π2n_{2} = \frac{3 \pi}{2}
Numerical solution
n1=42.4115008234622n_{1} = 42.4115008234622
n2=54.9778714378214n_{2} = 54.9778714378214
n3=−86.3937979737193n_{3} = -86.3937979737193
n4=−98.9601685880785n_{4} = -98.9601685880785
n5=29.845130209103n_{5} = 29.845130209103
n6=−42.4115008234622n_{6} = -42.4115008234622
n7=89.5353906273091n_{7} = 89.5353906273091
n8=−95.8185759344887n_{8} = -95.8185759344887
n9=−64.4026493985908n_{9} = -64.4026493985908
n10=14.1371669411541n_{10} = 14.1371669411541
n11=−17.2787595947439n_{11} = -17.2787595947439
n12=48.6946861306418n_{12} = 48.6946861306418
n13=−48.6946861306418n_{13} = -48.6946861306418
n14=−67.5442420521806n_{14} = -67.5442420521806
n15=−32.9867228626928n_{15} = -32.9867228626928
n16=−80.1106126665397n_{16} = -80.1106126665397
n17=83.2522053201295n_{17} = 83.2522053201295
n18=1.5707963267949n_{18} = 1.5707963267949
n19=10.9955742875643n_{19} = 10.9955742875643
n20=347.145988221672n_{20} = 347.145988221672
n21=−76.9690200129499n_{21} = -76.9690200129499
n22=−7.85398163397448n_{22} = -7.85398163397448
n23=98.9601685880785n_{23} = 98.9601685880785
n24=−4.71238898038469n_{24} = -4.71238898038469
n25=36.1283155162826n_{25} = 36.1283155162826
n26=20.4203522483337n_{26} = 20.4203522483337
n27=23.5619449019235n_{27} = 23.5619449019235
n28=51.8362787842316n_{28} = 51.8362787842316
n29=−45.553093477052n_{29} = -45.553093477052
n30=45.553093477052n_{30} = 45.553093477052
n31=−1.5707963267949n_{31} = -1.5707963267949
n32=−10.9955742875643n_{32} = -10.9955742875643
n33=26.7035375555132n_{33} = 26.7035375555132
n34=67.5442420521806n_{34} = 67.5442420521806
n35=92.6769832808989n_{35} = 92.6769832808989
n36=−58.1194640914112n_{36} = -58.1194640914112
n37=73.8274273593601n_{37} = 73.8274273593601
n38=−39.2699081698724n_{38} = -39.2699081698724
n39=95.8185759344887n_{39} = 95.8185759344887
n40=−23.5619449019235n_{40} = -23.5619449019235
n41=−70.6858347057703n_{41} = -70.6858347057703
n42=80.1106126665397n_{42} = 80.1106126665397
n43=58.1194640914112n_{43} = 58.1194640914112
n44=−14.1371669411541n_{44} = -14.1371669411541
n45=32.9867228626928n_{45} = 32.9867228626928
n46=−83.2522053201295n_{46} = -83.2522053201295
n47=7.85398163397448n_{47} = 7.85398163397448
n48=−89.5353906273091n_{48} = -89.5353906273091
n49=−29.845130209103n_{49} = -29.845130209103
n50=76.9690200129499n_{50} = 76.9690200129499
n51=86.3937979737193n_{51} = 86.3937979737193
n52=199.491133502952n_{52} = 199.491133502952
n53=−26.7035375555132n_{53} = -26.7035375555132
n54=−36.1283155162826n_{54} = -36.1283155162826
n55=−92.6769832808989n_{55} = -92.6769832808989
n56=70.6858347057703n_{56} = 70.6858347057703
n57=1173.38485611579n_{57} = 1173.38485611579
n58=−51.8362787842316n_{58} = -51.8362787842316
n59=−73.8274273593601n_{59} = -73.8274273593601
n60=17.2787595947439n_{60} = 17.2787595947439
n61=64.4026493985908n_{61} = 64.4026493985908
n62=−20.4203522483337n_{62} = -20.4203522483337
n63=4.71238898038469n_{63} = 4.71238898038469
n64=−54.9778714378214n_{64} = -54.9778714378214
n65=−422.544211907827n_{65} = -422.544211907827
n66=39.2699081698724n_{66} = 39.2699081698724
n67=−61.261056745001n_{67} = -61.261056745001
n68=61.261056745001n_{68} = 61.261056745001
The points of intersection with the Y axis coordinate
The graph crosses Y axis when n equals 0:
substitute n = 0 to cos(n)/n.
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
ddnf(n)=0\frac{d}{d n} f{\left(n \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddnf(n)=\frac{d}{d n} f{\left(n \right)} =
the first derivative
−sin⁡(n)n−cos⁡(n)n2=0- \frac{\sin{\left(n \right)}}{n} - \frac{\cos{\left(n \right)}}{n^{2}} = 0
Solve this equation
The roots of this equation
n1=−72.2427897046973n_{1} = -72.2427897046973
n2=78.5270825679419n_{2} = 78.5270825679419
n3=−18.7964043662102n_{3} = -18.7964043662102
n4=84.811211299318n_{4} = 84.811211299318
n5=−40.8162093266346n_{5} = -40.8162093266346
n6=43.9595528888955n_{6} = 43.9595528888955
n7=12.4864543952238n_{7} = 12.4864543952238
n8=−69.100567727981n_{8} = -69.100567727981
n9=18.7964043662102n_{9} = 18.7964043662102
n10=56.5309801938186n_{10} = 56.5309801938186
n11=2.79838604578389n_{11} = 2.79838604578389
n12=−15.644128370333n_{12} = -15.644128370333
n13=−37.672573565113n_{13} = -37.672573565113
n14=−100.521017074687n_{14} = -100.521017074687
n15=197.91528455229n_{15} = 197.91528455229
n16=−59.6735041304405n_{16} = -59.6735041304405
n17=72.2427897046973n_{17} = 72.2427897046973
n18=75.3849592185347n_{18} = 75.3849592185347
n19=28.2389365752603n_{19} = 28.2389365752603
n20=−25.0929104121121n_{20} = -25.0929104121121
n21=−31.3840740178899n_{21} = -31.3840740178899
n22=−75.3849592185347n_{22} = -75.3849592185347
n23=94.2371684817036n_{23} = 94.2371684817036
n24=37.672573565113n_{24} = 37.672573565113
n25=100.521017074687n_{25} = 100.521017074687
n26=40.8162093266346n_{26} = 40.8162093266346
n27=−28.2389365752603n_{27} = -28.2389365752603
n28=−6.12125046689807n_{28} = -6.12125046689807
n29=50.2455828375744n_{29} = 50.2455828375744
n30=34.5285657554621n_{30} = 34.5285657554621
n31=−169.640108529775n_{31} = -169.640108529775
n32=−97.3791034786112n_{32} = -97.3791034786112
n33=−56.5309801938186n_{33} = -56.5309801938186
n34=−34.5285657554621n_{34} = -34.5285657554621
n35=97.3791034786112n_{35} = 97.3791034786112
n36=−94.2371684817036n_{36} = -94.2371684817036
n37=81.6691650818489n_{37} = 81.6691650818489
n38=−78.5270825679419n_{38} = -78.5270825679419
n39=21.945612879981n_{39} = 21.945612879981
n40=−65.9582857893902n_{40} = -65.9582857893902
n41=15.644128370333n_{41} = 15.644128370333
n42=62.8159348889734n_{42} = 62.8159348889734
n43=−84.811211299318n_{43} = -84.811211299318
n44=6.12125046689807n_{44} = 6.12125046689807
n45=53.3883466217256n_{45} = 53.3883466217256
n46=−109.946647805931n_{46} = -109.946647805931
n47=−12.4864543952238n_{47} = -12.4864543952238
n48=−135.08108127842n_{48} = -135.08108127842
n49=−87.9532251106725n_{49} = -87.9532251106725
n50=−81.6691650818489n_{50} = -81.6691650818489
n51=31.3840740178899n_{51} = 31.3840740178899
n52=87.9532251106725n_{52} = 87.9532251106725
n53=−91.0952098694071n_{53} = -91.0952098694071
n54=−2.79838604578389n_{54} = -2.79838604578389
n55=−21.945612879981n_{55} = -21.945612879981
n56=−9.31786646179107n_{56} = -9.31786646179107
n57=−47.1026627703624n_{57} = -47.1026627703624
n58=−50.2455828375744n_{58} = -50.2455828375744
n59=69.100567727981n_{59} = 69.100567727981
n60=−62.8159348889734n_{60} = -62.8159348889734
n61=−53.3883466217256n_{61} = -53.3883466217256
n62=65.9582857893902n_{62} = 65.9582857893902
n63=−43.9595528888955n_{63} = -43.9595528888955
n64=9.31786646179107n_{64} = 9.31786646179107
n65=59.6735041304405n_{65} = 59.6735041304405
n66=25.0929104121121n_{66} = 25.0929104121121
n67=91.0952098694071n_{67} = 91.0952098694071
n68=47.1026627703624n_{68} = 47.1026627703624
The values of the extrema at the points:
(-72.24278970469729, 0.0138408859131547)

(78.52708256794193, -0.0127334276777468)

(-18.796404366210158, -0.0531265325613881)

(84.81121129931802, -0.0117900744410766)

(-40.81620932663458, 0.0244927205346957)

(43.959552888895495, 0.0227423004725314)

(12.486454395223781, 0.0798311807800032)

(-69.10056772798097, -0.0144701459746764)

(18.796404366210158, 0.0531265325613881)

(56.53098019381864, 0.0176866485521696)

(2.798386045783887, -0.336508416918395)

(-15.644128370333028, 0.0637915530395936)

(-37.67257356511297, -0.0265351630103045)

(-100.52101707468658, -0.00994767611536293)

(197.91528455229027, -0.00505260236866135)

(-59.67350413044053, 0.0167555036571887)

(72.24278970469729, -0.0138408859131547)

(75.38495921853475, 0.0132640786518247)

(28.238936575260272, -0.0353899155541688)

(-25.092910412112097, -0.0398202855500511)

(-31.38407401788986, -0.0318471321112693)

(-75.38495921853475, -0.0132640786518247)

(94.23716848170359, 0.01061092686295)

(37.67257356511297, 0.0265351630103045)

(100.52101707468658, 0.00994767611536293)

(40.81620932663458, -0.0244927205346957)

(-28.238936575260272, 0.0353899155541688)

(-6.1212504668980685, -0.161228034325064)

(50.24558283757444, 0.0198983065303553)

(34.52856575546206, -0.0289493889114503)

(-169.6401085297751, -0.00589472993500857)

(-97.3791034786112, 0.0102686022030809)

(-56.53098019381864, -0.0176866485521696)

(-34.52856575546206, 0.0289493889114503)

(97.3791034786112, -0.0102686022030809)

(-94.23716848170359, -0.01061092686295)

(81.66916508184887, 0.0122436055670467)

(-78.52708256794193, 0.0127334276777468)

(21.945612879981045, -0.0455199604051285)

(-65.95828578939016, 0.0151593553168405)

(15.644128370333028, -0.0637915530395936)

(62.81593488897342, 0.015917510583426)

(-84.81121129931802, 0.0117900744410766)

(6.1212504668980685, 0.161228034325064)

(53.38834662172563, -0.0187273944640866)

(-109.94664780593057, 0.00909494432157336)

(-12.486454395223781, -0.0798311807800032)

(-135.0810812784199, 0.00740275832666827)

(-87.95322511067255, -0.0113689449158811)

(-81.66916508184887, -0.0122436055670467)

(31.38407401788986, 0.0318471321112693)

(87.95322511067255, 0.0113689449158811)

(-91.09520986940714, 0.0109768642483425)

(-2.798386045783887, 0.336508416918395)

(-21.945612879981045, 0.0455199604051285)

(-9.317866461791066, 0.106707947715237)

(-47.10266277036235, 0.0212254394164143)

(-50.24558283757444, -0.0198983065303553)

(69.10056772798097, 0.0144701459746764)

(-62.81593488897342, -0.015917510583426)

(-53.38834662172563, 0.0187273944640866)

(65.95828578939016, -0.0151593553168405)

(-43.959552888895495, -0.0227423004725314)

(9.317866461791066, -0.106707947715237)

(59.67350413044053, -0.0167555036571887)

(25.092910412112097, 0.0398202855500511)

(91.09520986940714, -0.0109768642483425)

(47.10266277036235, -0.0212254394164143)


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:
n1=78.5270825679419n_{1} = 78.5270825679419
n2=−18.7964043662102n_{2} = -18.7964043662102
n3=84.811211299318n_{3} = 84.811211299318
n4=−69.100567727981n_{4} = -69.100567727981
n5=2.79838604578389n_{5} = 2.79838604578389
n6=−37.672573565113n_{6} = -37.672573565113
n7=−100.521017074687n_{7} = -100.521017074687
n8=197.91528455229n_{8} = 197.91528455229
n9=72.2427897046973n_{9} = 72.2427897046973
n10=28.2389365752603n_{10} = 28.2389365752603
n11=−25.0929104121121n_{11} = -25.0929104121121
n12=−31.3840740178899n_{12} = -31.3840740178899
n13=−75.3849592185347n_{13} = -75.3849592185347
n14=40.8162093266346n_{14} = 40.8162093266346
n15=−6.12125046689807n_{15} = -6.12125046689807
n16=34.5285657554621n_{16} = 34.5285657554621
n17=−169.640108529775n_{17} = -169.640108529775
n18=−56.5309801938186n_{18} = -56.5309801938186
n19=97.3791034786112n_{19} = 97.3791034786112
n20=−94.2371684817036n_{20} = -94.2371684817036
n21=21.945612879981n_{21} = 21.945612879981
n22=15.644128370333n_{22} = 15.644128370333
n23=53.3883466217256n_{23} = 53.3883466217256
n24=−12.4864543952238n_{24} = -12.4864543952238
n25=−87.9532251106725n_{25} = -87.9532251106725
n26=−81.6691650818489n_{26} = -81.6691650818489
n27=−50.2455828375744n_{27} = -50.2455828375744
n28=−62.8159348889734n_{28} = -62.8159348889734
n29=65.9582857893902n_{29} = 65.9582857893902
n30=−43.9595528888955n_{30} = -43.9595528888955
n31=9.31786646179107n_{31} = 9.31786646179107
n32=59.6735041304405n_{32} = 59.6735041304405
n33=91.0952098694071n_{33} = 91.0952098694071
n34=47.1026627703624n_{34} = 47.1026627703624
Maxima of the function at points:
n34=−72.2427897046973n_{34} = -72.2427897046973
n34=−40.8162093266346n_{34} = -40.8162093266346
n34=43.9595528888955n_{34} = 43.9595528888955
n34=12.4864543952238n_{34} = 12.4864543952238
n34=18.7964043662102n_{34} = 18.7964043662102
n34=56.5309801938186n_{34} = 56.5309801938186
n34=−15.644128370333n_{34} = -15.644128370333
n34=−59.6735041304405n_{34} = -59.6735041304405
n34=75.3849592185347n_{34} = 75.3849592185347
n34=94.2371684817036n_{34} = 94.2371684817036
n34=37.672573565113n_{34} = 37.672573565113
n34=100.521017074687n_{34} = 100.521017074687
n34=−28.2389365752603n_{34} = -28.2389365752603
n34=50.2455828375744n_{34} = 50.2455828375744
n34=−97.3791034786112n_{34} = -97.3791034786112
n34=−34.5285657554621n_{34} = -34.5285657554621
n34=81.6691650818489n_{34} = 81.6691650818489
n34=−78.5270825679419n_{34} = -78.5270825679419
n34=−65.9582857893902n_{34} = -65.9582857893902
n34=62.8159348889734n_{34} = 62.8159348889734
n34=−84.811211299318n_{34} = -84.811211299318
n34=6.12125046689807n_{34} = 6.12125046689807
n34=−109.946647805931n_{34} = -109.946647805931
n34=−135.08108127842n_{34} = -135.08108127842
n34=31.3840740178899n_{34} = 31.3840740178899
n34=87.9532251106725n_{34} = 87.9532251106725
n34=−91.0952098694071n_{34} = -91.0952098694071
n34=−2.79838604578389n_{34} = -2.79838604578389
n34=−21.945612879981n_{34} = -21.945612879981
n34=−9.31786646179107n_{34} = -9.31786646179107
n34=−47.1026627703624n_{34} = -47.1026627703624
n34=69.100567727981n_{34} = 69.100567727981
n34=−53.3883466217256n_{34} = -53.3883466217256
n34=25.0929104121121n_{34} = 25.0929104121121
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
d2dn2f(n)=0\frac{d^{2}}{d n^{2}} f{\left(n \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dn2f(n)=\frac{d^{2}}{d n^{2}} f{\left(n \right)} =
the second derivative
−cos⁡(n)+2sin⁡(n)n+2cos⁡(n)n2n=0\frac{- \cos{\left(n \right)} + \frac{2 \sin{\left(n \right)}}{n} + \frac{2 \cos{\left(n \right)}}{n^{2}}}{n} = 0
Solve this equation
The roots of this equation
n1=61.2283863503723n_{1} = 61.2283863503723
n2=7.5873993379941n_{2} = 7.5873993379941
n3=−98.9399529307048n_{3} = -98.9399529307048
n4=−271.740404503579n_{4} = -271.740404503579
n5=−76.9430238267933n_{5} = -76.9430238267933
n6=−51.7976574095537n_{6} = -51.7976574095537
n7=23.4766510546492n_{7} = 23.4766510546492
n8=−89.5130456566371n_{8} = -89.5130456566371
n9=36.0728437679879n_{9} = 36.0728437679879
n10=45.5091321154553n_{10} = 45.5091321154553
n11=10.8095072981602n_{11} = 10.8095072981602
n12=−48.6535676048409n_{12} = -48.6535676048409
n13=−7.5873993379941n_{13} = -7.5873993379941
n14=4.2222763997912n_{14} = 4.2222763997912
n15=92.655396245836n_{15} = 92.655396245836
n16=51.7976574095537n_{16} = 51.7976574095537
n17=−26.6283591640252n_{17} = -26.6283591640252
n18=−29.7779159141436n_{18} = -29.7779159141436
n19=−13.9937625671267n_{19} = -13.9937625671267
n20=−23.4766510546492n_{20} = -23.4766510546492
n21=−80.0856368040887n_{21} = -80.0856368040887
n22=−20.3217772482235n_{22} = -20.3217772482235
n23=13.9937625671267n_{23} = 13.9937625671267
n24=−17.1619600917303n_{24} = -17.1619600917303
n25=42.3642737086586n_{25} = 42.3642737086586
n26=73.8003238908837n_{26} = 73.8003238908837
n27=−70.6575253785884n_{27} = -70.6575253785884
n28=83.2281726832512n_{28} = 83.2281726832512
n29=−95.7976970894915n_{29} = -95.7976970894915
n30=95.7976970894915n_{30} = 95.7976970894915
n31=89.5130456566371n_{31} = 89.5130456566371
n32=54.9414610202918n_{32} = 54.9414610202918
n33=17.1619600917303n_{33} = 17.1619600917303
n34=67.5146145048817n_{34} = 67.5146145048817
n35=−58.085025007445n_{35} = -58.085025007445
n36=98.9399529307048n_{36} = 98.9399529307048
n37=−67.5146145048817n_{37} = -67.5146145048817
n38=230.898398112111n_{38} = 230.898398112111
n39=−39.218890250481n_{39} = -39.218890250481
n40=−45.5091321154553n_{40} = -45.5091321154553
n41=64.3715747870554n_{41} = 64.3715747870554
n42=39.218890250481n_{42} = 39.218890250481
n43=29.7779159141436n_{43} = 29.7779159141436
n44=−4.2222763997912n_{44} = -4.2222763997912
n45=−42.3642737086586n_{45} = -42.3642737086586
n46=−92.655396245836n_{46} = -92.655396245836
n47=70.6575253785884n_{47} = 70.6575253785884
n48=86.370639887736n_{48} = 86.370639887736
n49=48.6535676048409n_{49} = 48.6535676048409
n50=20.3217772482235n_{50} = 20.3217772482235
n51=−36.0728437679879n_{51} = -36.0728437679879
n52=−73.8003238908837n_{52} = -73.8003238908837
n53=−64.3715747870554n_{53} = -64.3715747870554
n54=80.0856368040887n_{54} = 80.0856368040887
n55=−32.9259431758392n_{55} = -32.9259431758392
n56=−86.370639887736n_{56} = -86.370639887736
n57=26.6283591640252n_{57} = 26.6283591640252
n58=−10.8095072981602n_{58} = -10.8095072981602
n59=−83.2281726832512n_{59} = -83.2281726832512
n60=−61.2283863503723n_{60} = -61.2283863503723
n61=−54.9414610202918n_{61} = -54.9414610202918
n62=58.085025007445n_{62} = 58.085025007445
n63=32.9259431758392n_{63} = 32.9259431758392
n64=76.9430238267933n_{64} = 76.9430238267933
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
n1=0n_{1} = 0

lim⁡n→0−(−cos⁡(n)+2sin⁡(n)n+2cos⁡(n)n2n)=−∞\lim_{n \to 0^-}\left(\frac{- \cos{\left(n \right)} + \frac{2 \sin{\left(n \right)}}{n} + \frac{2 \cos{\left(n \right)}}{n^{2}}}{n}\right) = -\infty
lim⁡n→0+(−cos⁡(n)+2sin⁡(n)n+2cos⁡(n)n2n)=∞\lim_{n \to 0^+}\left(\frac{- \cos{\left(n \right)} + \frac{2 \sin{\left(n \right)}}{n} + \frac{2 \cos{\left(n \right)}}{n^{2}}}{n}\right) = \infty
- the limits are not equal, so
n1=0n_{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:
n1=0n_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at n->+oo and n->-oo
lim⁡n→−∞(cos⁡(n)n)=0\lim_{n \to -\infty}\left(\frac{\cos{\left(n \right)}}{n}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡n→∞(cos⁡(n)n)=0\lim_{n \to \infty}\left(\frac{\cos{\left(n \right)}}{n}\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(n)/n, divided by n at n->+oo and n ->-oo
lim⁡n→−∞(cos⁡(n)n2)=0\lim_{n \to -\infty}\left(\frac{\cos{\left(n \right)}}{n^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡n→∞(cos⁡(n)n2)=0\lim_{n \to \infty}\left(\frac{\cos{\left(n \right)}}{n^{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(-n) и f = -f(-n).
So, check:
cos⁡(n)n=−cos⁡(n)n\frac{\cos{\left(n \right)}}{n} = - \frac{\cos{\left(n \right)}}{n}
- No
cos⁡(n)n=cos⁡(n)n\frac{\cos{\left(n \right)}}{n} = \frac{\cos{\left(n \right)}}{n}
- No
so, the function
not is
neither even, nor odd
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