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Plotting a function graph/ cos(x)/(x+1)

Graphing y = cos(x)/(x+1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(x)
f(x) = ------
       x + 1
f(x)=cos⁡(x)x+1f{\left(x \right)} = \frac{\cos{\left(x \right)}}{x + 1} 
f = cos(x)/(x + 1)
The graph of the function
02468-8-6-4-2-1010-2525
The domain of the function
The points at which the function is not precisely defined:
x1=−1x_{1} = -1
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+1=0\frac{\cos{\left(x \right)}}{x + 1} = 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=48.6946861306418x_{1} = 48.6946861306418
x2=92.6769832808989x_{2} = 92.6769832808989
x3=86.3937979737193x_{3} = 86.3937979737193
x4=−7.85398163397448x_{4} = -7.85398163397448
x5=−86.3937979737193x_{5} = -86.3937979737193
x6=1.5707963267949x_{6} = 1.5707963267949
x7=−64.4026493985908x_{7} = -64.4026493985908
x8=−58.1194640914112x_{8} = -58.1194640914112
x9=−83.2522053201295x_{9} = -83.2522053201295
x10=−161.792021659874x_{10} = -161.792021659874
x11=54.9778714378214x_{11} = 54.9778714378214
x12=89.5353906273091x_{12} = 89.5353906273091
x13=−54.9778714378214x_{13} = -54.9778714378214
x14=−20.4203522483337x_{14} = -20.4203522483337
x15=32.9867228626928x_{15} = 32.9867228626928
x16=−17.2787595947439x_{16} = -17.2787595947439
x17=23.5619449019235x_{17} = 23.5619449019235
x18=−256.039801267568x_{18} = -256.039801267568
x19=−45.553093477052x_{19} = -45.553093477052
x20=64.4026493985908x_{20} = 64.4026493985908
x21=45.553093477052x_{21} = 45.553093477052
x22=83.2522053201295x_{22} = 83.2522053201295
x23=−29.845130209103x_{23} = -29.845130209103
x24=−51.8362787842316x_{24} = -51.8362787842316
x25=230.90706003885x_{25} = 230.90706003885
x26=80.1106126665397x_{26} = 80.1106126665397
x27=−39.2699081698724x_{27} = -39.2699081698724
x28=−92.6769832808989x_{28} = -92.6769832808989
x29=4.71238898038469x_{29} = 4.71238898038469
x30=70.6858347057703x_{30} = 70.6858347057703
x31=36.1283155162826x_{31} = 36.1283155162826
x32=−70.6858347057703x_{32} = -70.6858347057703
x33=−48.6946861306418x_{33} = -48.6946861306418
x34=−127.234502470387x_{34} = -127.234502470387
x35=42.4115008234622x_{35} = 42.4115008234622
x36=−42.4115008234622x_{36} = -42.4115008234622
x37=−67.5442420521806x_{37} = -67.5442420521806
x38=10.9955742875643x_{38} = 10.9955742875643
x39=98.9601685880785x_{39} = 98.9601685880785
x40=−23.5619449019235x_{40} = -23.5619449019235
x41=20.4203522483337x_{41} = 20.4203522483337
x42=−61.261056745001x_{42} = -61.261056745001
x43=−10.9955742875643x_{43} = -10.9955742875643
x44=17.2787595947439x_{44} = 17.2787595947439
x45=−95.8185759344887x_{45} = -95.8185759344887
x46=−36.1283155162826x_{46} = -36.1283155162826
x47=61.261056745001x_{47} = 61.261056745001
x48=73.8274273593601x_{48} = 73.8274273593601
x49=14.1371669411541x_{49} = 14.1371669411541
x50=−26.7035375555132x_{50} = -26.7035375555132
x51=51.8362787842316x_{51} = 51.8362787842316
x52=−89.5353906273091x_{52} = -89.5353906273091
x53=39.2699081698724x_{53} = 39.2699081698724
x54=−32.9867228626928x_{54} = -32.9867228626928
x55=−14.1371669411541x_{55} = -14.1371669411541
x56=−4.71238898038469x_{56} = -4.71238898038469
x57=−76.9690200129499x_{57} = -76.9690200129499
x58=95.8185759344887x_{58} = 95.8185759344887
x59=76.9690200129499x_{59} = 76.9690200129499
x60=58.1194640914112x_{60} = 58.1194640914112
x61=−80.1106126665397x_{61} = -80.1106126665397
x62=−73.8274273593601x_{62} = -73.8274273593601
x63=7.85398163397448x_{63} = 7.85398163397448
x64=−1.5707963267949x_{64} = -1.5707963267949
x65=29.845130209103x_{65} = 29.845130209103
x66=67.5442420521806x_{66} = 67.5442420521806
x67=26.7035375555132x_{67} = 26.7035375555132
x68=−98.9601685880785x_{68} = -98.9601685880785
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 + 1).
cos⁡(0)1\frac{\cos{\left(0 \right)}}{1}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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+1−cos⁡(x)(x+1)2=0- \frac{\sin{\left(x \right)}}{x + 1} - \frac{\cos{\left(x \right)}}{\left(x + 1\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=9.32825706323943x_{1} = 9.32825706323943
x2=43.9600588531378x_{2} = 43.9600588531378
x3=−94.2370546693974x_{3} = -94.2370546693974
x4=2.88996969767843x_{4} = 2.88996969767843
x5=72.242978694986x_{5} = 72.242978694986
x6=18.7990914357831x_{6} = 18.7990914357831
x7=69.1007741687956x_{7} = 69.1007741687956
x8=−31.3830252979972x_{8} = -31.3830252979972
x9=−25.0912562079058x_{9} = -25.0912562079058
x10=−91.0950880256329x_{10} = -91.0950880256329
x11=−28.2376364595748x_{11} = -28.2376364595748
x12=−637.741738184573x_{12} = -637.741738184573
x13=−53.3879890840753x_{13} = -53.3879890840753
x14=40.8167952172419x_{14} = 40.8167952172419
x15=1313.18496827279x_{15} = 1313.18496827279
x16=−81.6690132946536x_{16} = -81.6690132946536
x17=15.6479679638982x_{17} = 15.6479679638982
x18=−72.24259540785x_{18} = -72.24259540785
x19=−2.57625015820118x_{19} = -2.57625015820118
x20=6.14411351301787x_{20} = 6.14411351301787
x21=−12.4794779911025x_{21} = -12.4794779911025
x22=94.2372799036618x_{22} = 94.2372799036618
x23=84.8113487041494x_{23} = 84.8113487041494
x24=81.6693131963402x_{24} = 81.6693131963402
x25=56.5312876685112x_{25} = 56.5312876685112
x26=−47.1022022669651x_{26} = -47.1022022669651
x27=100.521115065812x_{27} = 100.521115065812
x28=−100.520917114109x_{28} = -100.520917114109
x29=−62.815677356778x_{29} = -62.815677356778
x30=37.673259943911x_{30} = 37.673259943911
x31=−97.378996929011x_{31} = -97.378996929011
x32=47.1031041186137x_{32} = 47.1031041186137
x33=65.9585122146304x_{33} = 65.9585122146304
x34=78.5272426949571x_{34} = 78.5272426949571
x35=−87.9530943542027x_{35} = -87.9530943542027
x36=−15.6397620877646x_{36} = -15.6397620877646
x37=53.388691007263x_{37} = 53.388691007263
x38=−21.9434371567881x_{38} = -21.9434371567881
x39=−59.6732185170696x_{39} = -59.6732185170696
x40=12.492390025579x_{40} = 12.492390025579
x41=172.781841669816x_{41} = 172.781841669816
x42=50.2459712046114x_{42} = 50.2459712046114
x43=31.38505790634x_{43} = 31.38505790634
x44=−75.3847808857452x_{44} = -75.3847808857452
x45=34.5293808983144x_{45} = 34.5293808983144
x46=−37.6718497263809x_{46} = -37.6718497263809
x47=−34.527701946778x_{47} = -34.527701946778
x48=75.3851328811964x_{48} = 75.3851328811964
x49=−6.08916120309943x_{49} = -6.08916120309943
x50=−43.9590233567938x_{50} = -43.9590233567938
x51=59.6737803264459x_{51} = 59.6737803264459
x52=−69.1003552230555x_{52} = -69.1003552230555
x53=−40.8155939881502x_{53} = -40.8155939881502
x54=28.2401476526276x_{54} = 28.2401476526276
x55=−9.30494468339504x_{55} = -9.30494468339504
x56=−78.5269183093816x_{56} = -78.5269183093816
x57=21.9475985837942x_{57} = 21.9475985837942
x58=−18.7934144113698x_{58} = -18.7934144113698
x59=−56.5306616416093x_{59} = -56.5306616416093
x60=91.0953290668266x_{60} = 91.0953290668266
x61=87.9533529268738x_{61} = 87.9533529268738
x62=197.915309953386x_{62} = 197.915309953386
x63=−84.8110706151124x_{63} = -84.8110706151124
x64=25.0944376288815x_{64} = 25.0944376288815
x65=62.8161843480611x_{65} = 62.8161843480611
x66=−50.2451786914948x_{66} = -50.2451786914948
x67=−65.9580523911179x_{67} = -65.9580523911179
x68=97.379207861883x_{68} = 97.379207861883
The values of the extrema at the points:
(9.328257063239425, -0.0963710979823201)

(43.960058853137774, 0.022236464203186)

(-94.23705466939735, -0.010724732692878)

(2.8899696976784344, -0.248976134877405)

(72.242978694986, -0.0136519134817116)

(18.79909143578314, 0.0504430691319447)

(69.10077416879557, 0.0142637264671467)

(-31.38302529799723, -0.0328953023371544)

(-25.091256207905772, -0.0414731225016059)

(-91.09508802563293, 0.0110987005999837)

(-28.237636459574798, 0.0366891865463047)

(-637.7417381845734, 0.00157049350907233)

(-53.387989084075315, 0.0190848682073296)

(40.81679521724192, -0.023907001519389)

(1313.1849682727866, 0.000760927673528925)

(-81.66901329465364, -0.0123953812433342)

(15.647967963898166, -0.0599593189797558)

(-72.24259540785, 0.0140351638863266)

(-2.5762501582011796, 0.535705052303484)

(6.1441135130178655, 0.138623930394573)

(-12.479477991102517, -0.0867833198945747)

(94.23727990366179, 0.0104995111118831)

(84.81134870414938, -0.0116526790492257)

(81.66931319634023, 0.0120955020439642)

(56.53128766851124, 0.0173792211238612)

(-47.10220226696507, 0.0216858368023364)

(100.52111506581193, 0.00984968979094353)

(-100.52091711410945, -0.0100476316966419)

(-62.815677356778, -0.0161750096209984)

(37.673259943911006, 0.0258490197028825)

(-97.37899692901101, 0.0103751461271118)

(47.10310411861372, -0.0207841885412821)

(65.95851221463039, -0.0149329557083856)

(78.52724269495707, -0.0125733134820883)

(-87.95309435420273, -0.0114996928375307)

(-15.63976208776456, 0.0681483206400774)

(53.388691007263, -0.0183830682189117)

(-21.94343715678808, 0.0476933188520339)

(-59.673218517069586, 0.0170410762454831)

(12.492390025578958, 0.0739131230459364)

(172.781841669816, -0.0057542458670116)

(50.24597120461141, 0.0195100148956696)

(31.385057906339963, 0.0308637274812354)

(-75.38478088574516, -0.0134423955413013)

(34.5293808983144, -0.0281345781753277)

(-37.67184972638089, -0.0272587398500595)

(-34.52770194677802, 0.0298128246468963)

(75.38513288119637, 0.0130904310684593)

(-6.089161203099427, -0.192809042427521)

(-43.95902335679378, -0.0232716924030311)

(59.67378032644585, -0.0164793457895915)

(-69.1003552230555, -0.0146826283229769)

(-40.81559398815024, 0.0251078697468112)

(28.240147652627645, -0.0341795711715136)

(-9.304944683395044, 0.119546681963348)

(-78.5269183093816, 0.0128976727485698)

(21.947598583794207, -0.0435362264748061)

(-18.793414411369753, -0.0561120230339157)

(-56.53066164160934, -0.0180051500304447)

(91.09532906682657, -0.0108576739325778)

(87.9533529268738, 0.0112411368826843)

(197.91530995338616, -0.00502720159537522)

(-84.81107061511238, 0.0119307487512748)

(25.094437628881476, 0.0382942342355763)

(62.81618434806106, 0.0156680826074814)

(-50.245178691494786, -0.0203023709567303)

(-65.9580523911179, 0.0153927263543733)

(97.37920786188297, -0.0101642243790071)


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=9.32825706323943x_{1} = 9.32825706323943
x2=−94.2370546693974x_{2} = -94.2370546693974
x3=2.88996969767843x_{3} = 2.88996969767843
x4=72.242978694986x_{4} = 72.242978694986
x5=−31.3830252979972x_{5} = -31.3830252979972
x6=−25.0912562079058x_{6} = -25.0912562079058
x7=40.8167952172419x_{7} = 40.8167952172419
x8=−81.6690132946536x_{8} = -81.6690132946536
x9=15.6479679638982x_{9} = 15.6479679638982
x10=−12.4794779911025x_{10} = -12.4794779911025
x11=84.8113487041494x_{11} = 84.8113487041494
x12=−100.520917114109x_{12} = -100.520917114109
x13=−62.815677356778x_{13} = -62.815677356778
x14=47.1031041186137x_{14} = 47.1031041186137
x15=65.9585122146304x_{15} = 65.9585122146304
x16=78.5272426949571x_{16} = 78.5272426949571
x17=−87.9530943542027x_{17} = -87.9530943542027
x18=53.388691007263x_{18} = 53.388691007263
x19=172.781841669816x_{19} = 172.781841669816
x20=−75.3847808857452x_{20} = -75.3847808857452
x21=34.5293808983144x_{21} = 34.5293808983144
x22=−37.6718497263809x_{22} = -37.6718497263809
x23=−6.08916120309943x_{23} = -6.08916120309943
x24=−43.9590233567938x_{24} = -43.9590233567938
x25=59.6737803264459x_{25} = 59.6737803264459
x26=−69.1003552230555x_{26} = -69.1003552230555
x27=28.2401476526276x_{27} = 28.2401476526276
x28=21.9475985837942x_{28} = 21.9475985837942
x29=−18.7934144113698x_{29} = -18.7934144113698
x30=−56.5306616416093x_{30} = -56.5306616416093
x31=91.0953290668266x_{31} = 91.0953290668266
x32=197.915309953386x_{32} = 197.915309953386
x33=−50.2451786914948x_{33} = -50.2451786914948
x34=97.379207861883x_{34} = 97.379207861883
Maxima of the function at points:
x34=43.9600588531378x_{34} = 43.9600588531378
x34=18.7990914357831x_{34} = 18.7990914357831
x34=69.1007741687956x_{34} = 69.1007741687956
x34=−91.0950880256329x_{34} = -91.0950880256329
x34=−28.2376364595748x_{34} = -28.2376364595748
x34=−637.741738184573x_{34} = -637.741738184573
x34=−53.3879890840753x_{34} = -53.3879890840753
x34=1313.18496827279x_{34} = 1313.18496827279
x34=−72.24259540785x_{34} = -72.24259540785
x34=−2.57625015820118x_{34} = -2.57625015820118
x34=6.14411351301787x_{34} = 6.14411351301787
x34=94.2372799036618x_{34} = 94.2372799036618
x34=81.6693131963402x_{34} = 81.6693131963402
x34=56.5312876685112x_{34} = 56.5312876685112
x34=−47.1022022669651x_{34} = -47.1022022669651
x34=100.521115065812x_{34} = 100.521115065812
x34=37.673259943911x_{34} = 37.673259943911
x34=−97.378996929011x_{34} = -97.378996929011
x34=−15.6397620877646x_{34} = -15.6397620877646
x34=−21.9434371567881x_{34} = -21.9434371567881
x34=−59.6732185170696x_{34} = -59.6732185170696
x34=12.492390025579x_{34} = 12.492390025579
x34=50.2459712046114x_{34} = 50.2459712046114
x34=31.38505790634x_{34} = 31.38505790634
x34=−34.527701946778x_{34} = -34.527701946778
x34=75.3851328811964x_{34} = 75.3851328811964
x34=−40.8155939881502x_{34} = -40.8155939881502
x34=−9.30494468339504x_{34} = -9.30494468339504
x34=−78.5269183093816x_{34} = -78.5269183093816
x34=87.9533529268738x_{34} = 87.9533529268738
x34=−84.8110706151124x_{34} = -84.8110706151124
x34=25.0944376288815x_{34} = 25.0944376288815
x34=62.8161843480611x_{34} = 62.8161843480611
x34=−65.9580523911179x_{34} = -65.9580523911179
Decreasing at intervals
[197.915309953386,∞)\left[197.915309953386, \infty\right)
Increasing at intervals
(−∞,−100.520917114109]\left(-\infty, -100.520917114109\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+1+2cos⁡(x)(x+1)2x+1=0\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1} = 0
Solve this equation
The roots of this equation
x1=−20.3166301288662x_{1} = -20.3166301288662
x2=−45.5081427660817x_{2} = -45.5081427660817
x3=29.7801075137773x_{3} = 29.7801075137773
x4=98.9401552763972x_{4} = 98.9401552763972
x5=−48.6527034788051x_{5} = -48.6527034788051
x6=80.0859449790141x_{6} = 80.0859449790141
x7=45.5100787997204x_{7} = 45.5100787997204
x8=32.9277399444348x_{8} = 32.9277399444348
x9=−64.3710840254309x_{9} = -64.3710840254309
x10=92.6556268279389x_{10} = 92.6556268279389
x11=−54.9407852505616x_{11} = -54.9407852505616
x12=−89.5127931011103x_{12} = -89.5127931011103
x13=89.5132926274963x_{13} = 89.5132926274963
x14=−98.9397464504172x_{14} = -98.9397464504172
x15=−39.2175523279643x_{15} = -39.2175523279643
x16=4.32863617605124x_{16} = 4.32863617605124
x17=−83.2278802717944x_{17} = -83.2278802717944
x18=136.64474976163x_{18} = 136.64474976163
x19=−70.6571186927646x_{19} = -70.6571186927646
x20=73.80068645168x_{20} = 73.80068645168
x21=−76.9426813176863x_{21} = -76.9426813176863
x22=102.082358062481x_{22} = 102.082358062481
x23=−58.0844210975337x_{23} = -58.0844210975337
x24=−29.7755709323142x_{24} = -29.7755709323142
x25=−92.6551606286758x_{25} = -92.6551606286758
x26=17.1684571899007x_{26} = 17.1684571899007
x27=54.9421125829153x_{27} = 54.9421125829153
x28=−26.6254109350763x_{28} = -26.6254109350763
x29=67.5150472396589x_{29} = 67.5150472396589
x30=61.2289118119026x_{30} = 61.2289118119026
x31=−7.54372449628009x_{31} = -7.54372449628009
x32=−80.0853208283276x_{32} = -80.0853208283276
x33=−61.2278434114583x_{33} = -61.2278434114583
x34=86.3709050594723x_{34} = 86.3709050594723
x35=−4.00507341668955x_{35} = -4.00507341668955
x36=−95.7974767616183x_{36} = -95.7974767616183
x37=48.654396838104x_{37} = 48.654396838104
x38=−10.7898786754269x_{38} = -10.7898786754269
x39=95.797912862081x_{39} = 95.797912862081
x40=−51.7968961320869x_{40} = -51.7968961320869
x41=−86.3703684986956x_{41} = -86.3703684986956
x42=76.9433575383977x_{42} = 76.9433575383977
x43=7.61991323310644x_{43} = 7.61991323310644
x44=70.657920700132x_{44} = 70.657920700132
x45=14.0034717913284x_{45} = 14.0034717913284
x46=83.228458145445x_{46} = 83.228458145445
x47=23.4801553706306x_{47} = 23.4801553706306
x48=39.22016138731x_{48} = 39.22016138731
x49=58.0856084395179x_{49} = 58.0856084395179
x50=10.825651157762x_{50} = 10.825651157762
x51=42.3653647291314x_{51} = 42.3653647291314
x52=51.7983897861238x_{52} = 51.7983897861238
x53=20.3264348242219x_{53} = 20.3264348242219
x54=−17.1546413657741x_{54} = -17.1546413657741
x55=36.0743437126941x_{55} = 36.0743437126941
x56=−32.9240332040206x_{56} = -32.9240332040206
x57=−73.7999513585394x_{57} = -73.7999513585394
x58=−23.4728313498836x_{58} = -23.4728313498836
x59=−67.5141687409854x_{59} = -67.5141687409854
x60=−36.0712578833702x_{60} = -36.0712578833702
x61=64.3720505127272x_{61} = 64.3720505127272
x62=−13.9825085391948x_{62} = -13.9825085391948
x63=26.6310922236611x_{63} = 26.6310922236611
x64=−42.3631297553676x_{64} = -42.3631297553676
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=−1x_{1} = -1

lim⁡x→−1−(−cos⁡(x)+2sin⁡(x)x+1+2cos⁡(x)(x+1)2x+1)=−∞\lim_{x \to -1^-}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1}\right) = -\infty
lim⁡x→−1+(−cos⁡(x)+2sin⁡(x)x+1+2cos⁡(x)(x+1)2x+1)=∞\lim_{x \to -1^+}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x + 1} + \frac{2 \cos{\left(x \right)}}{\left(x + 1\right)^{2}}}{x + 1}\right) = \infty
- the limits are not equal, so
x1=−1x_{1} = -1
- 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
[102.082358062481,∞)\left[102.082358062481, \infty\right)
Convex at the intervals
(−∞,−95.7974767616183]\left(-\infty, -95.7974767616183\right]
Vertical asymptotes
Have:
x1=−1x_{1} = -1
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+1)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x + 1}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(cos⁡(x)x+1)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x + 1}\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 + 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x)x(x+1))=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x \left(x + 1\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x)x(x+1))=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x \left(x + 1\right)}\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+1=cos⁡(x)1−x\frac{\cos{\left(x \right)}}{x + 1} = \frac{\cos{\left(x \right)}}{1 - x}
- No
cos⁡(x)x+1=−cos⁡(x)1−x\frac{\cos{\left(x \right)}}{x + 1} = - \frac{\cos{\left(x \right)}}{1 - x}
- No
so, the function
not is
neither even, nor odd
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