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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=−714.712328691678x_{6} = -714.712328691678
x7=−64.4026493985908x_{7} = -64.4026493985908
x8=−58.1194640914112x_{8} = -58.1194640914112
x9=−83.2522053201295x_{9} = -83.2522053201295
x10=−54.9778714378214x_{10} = -54.9778714378214
x11=54.9778714378214x_{11} = 54.9778714378214
x12=89.5353906273091x_{12} = 89.5353906273091
x13=−20.4203522483337x_{13} = -20.4203522483337
x14=32.9867228626928x_{14} = 32.9867228626928
x15=−17.2787595947439x_{15} = -17.2787595947439
x16=391.128285371929x_{16} = 391.128285371929
x17=23.5619449019235x_{17} = 23.5619449019235
x18=−45.553093477052x_{18} = -45.553093477052
x19=64.4026493985908x_{19} = 64.4026493985908
x20=45.553093477052x_{20} = 45.553093477052
x21=83.2522053201295x_{21} = 83.2522053201295
x22=−29.845130209103x_{22} = -29.845130209103
x23=−51.8362787842316x_{23} = -51.8362787842316
x24=80.1106126665397x_{24} = 80.1106126665397
x25=−39.2699081698724x_{25} = -39.2699081698724
x26=−92.6769832808989x_{26} = -92.6769832808989
x27=199.491133502952x_{27} = 199.491133502952
x28=26.7035375555132x_{28} = 26.7035375555132
x29=4.71238898038469x_{29} = 4.71238898038469
x30=70.6858347057703x_{30} = 70.6858347057703
x31=1173.38485611579x_{31} = 1173.38485611579
x32=36.1283155162826x_{32} = 36.1283155162826
x33=−70.6858347057703x_{33} = -70.6858347057703
x34=−48.6946861306418x_{34} = -48.6946861306418
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=−136.659280431156x_{55} = -136.659280431156
x56=−4.71238898038469x_{56} = -4.71238898038469
x57=−14.1371669411541x_{57} = -14.1371669411541
x58=−76.9690200129499x_{58} = -76.9690200129499
x59=95.8185759344887x_{59} = 95.8185759344887
x60=76.9690200129499x_{60} = 76.9690200129499
x61=58.1194640914112x_{61} = 58.1194640914112
x62=−80.1106126665397x_{62} = -80.1106126665397
x63=−73.8274273593601x_{63} = -73.8274273593601
x64=7.85398163397448x_{64} = 7.85398163397448
x65=−1.5707963267949x_{65} = -1.5707963267949
x66=29.845130209103x_{66} = 29.845130209103
x67=67.5442420521806x_{67} = 67.5442420521806
x68=538.78314009065x_{68} = 538.78314009065
x69=−98.9601685880785x_{69} = -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=100.520917114109x_{1} = 100.520917114109
x2=−25.0944376288815x_{2} = -25.0944376288815
x3=2.57625015820118x_{3} = 2.57625015820118
x4=−37.673259943911x_{4} = -37.673259943911
x5=191.631906205502x_{5} = 191.631906205502
x6=15.6397620877646x_{6} = 15.6397620877646
x7=188.490225654589x_{7} = 188.490225654589
x8=56.5306616416093x_{8} = 56.5306616416093
x9=−69.1007741687956x_{9} = -69.1007741687956
x10=91.0950880256329x_{10} = 91.0950880256329
x11=87.9530943542027x_{11} = 87.9530943542027
x12=−15.6479679638982x_{12} = -15.6479679638982
x13=50.2451786914948x_{13} = 50.2451786914948
x14=−6.14411351301787x_{14} = -6.14411351301787
x15=−12.492390025579x_{15} = -12.492390025579
x16=−53.388691007263x_{16} = -53.388691007263
x17=−94.2372799036618x_{17} = -94.2372799036618
x18=28.2376364595748x_{18} = 28.2376364595748
x19=−2.88996969767843x_{19} = -2.88996969767843
x20=−59.6737803264459x_{20} = -59.6737803264459
x21=−1077.56535302402x_{21} = -1077.56535302402
x22=34.527701946778x_{22} = 34.527701946778
x23=31.3830252979972x_{23} = 31.3830252979972
x24=9.30494468339504x_{24} = 9.30494468339504
x25=−100.521115065812x_{25} = -100.521115065812
x26=12.4794779911025x_{26} = 12.4794779911025
x27=21.9434371567881x_{27} = 21.9434371567881
x28=75.3847808857452x_{28} = 75.3847808857452
x29=−78.5272426949571x_{29} = -78.5272426949571
x30=94.2370546693974x_{30} = 94.2370546693974
x31=81.6690132946536x_{31} = 81.6690132946536
x32=−72.242978694986x_{32} = -72.242978694986
x33=−122.514017419839x_{33} = -122.514017419839
x34=25.0912562079058x_{34} = 25.0912562079058
x35=69.1003552230555x_{35} = 69.1003552230555
x36=43.9590233567938x_{36} = 43.9590233567938
x37=−50.2459712046114x_{37} = -50.2459712046114
x38=78.5269183093816x_{38} = 78.5269183093816
x39=40.8155939881502x_{39} = 40.8155939881502
x40=59.6732185170696x_{40} = 59.6732185170696
x41=65.9580523911179x_{41} = 65.9580523911179
x42=72.24259540785x_{42} = 72.24259540785
x43=37.6718497263809x_{43} = 37.6718497263809
x44=18.7934144113698x_{44} = 18.7934144113698
x45=−62.8161843480611x_{45} = -62.8161843480611
x46=−43.9600588531378x_{46} = -43.9600588531378
x47=−97.379207861883x_{47} = -97.379207861883
x48=−28.2401476526276x_{48} = -28.2401476526276
x49=−87.9533529268738x_{49} = -87.9533529268738
x50=−75.3851328811964x_{50} = -75.3851328811964
x51=−546.635295693876x_{51} = -546.635295693876
x52=−18.7990914357831x_{52} = -18.7990914357831
x53=62.815677356778x_{53} = 62.815677356778
x54=−84.8113487041494x_{54} = -84.8113487041494
x55=84.8110706151124x_{55} = 84.8110706151124
x56=6.08916120309943x_{56} = 6.08916120309943
x57=53.3879890840753x_{57} = 53.3879890840753
x58=−47.1031041186137x_{58} = -47.1031041186137
x59=47.1022022669651x_{59} = 47.1022022669651
x60=−21.9475985837942x_{60} = -21.9475985837942
x61=−65.9585122146304x_{61} = -65.9585122146304
x62=−9.32825706323943x_{62} = -9.32825706323943
x63=−34.5293808983144x_{63} = -34.5293808983144
x64=−81.6693131963402x_{64} = -81.6693131963402
x65=−40.8167952172419x_{65} = -40.8167952172419
x66=97.378996929011x_{66} = 97.378996929011
x67=−31.38505790634x_{67} = -31.38505790634
x68=−56.5312876685112x_{68} = -56.5312876685112
x69=−91.0953290668266x_{69} = -91.0953290668266
The values of the extrema at the points:
(100.52091711410945, 0.0100476316966419)

(-25.094437628881476, -0.0382942342355763)

(2.5762501582011796, -0.535705052303484)

(-37.673259943911006, -0.0258490197028825)

(191.63190620550185, -0.00524563941809883)

(15.63976208776456, -0.0681483206400774)

(188.4902256545889, 0.00533353551155559)

(56.53066164160934, 0.0180051500304447)

(-69.10077416879557, -0.0142637264671467)

(91.09508802563293, -0.0110987005999837)

(87.95309435420273, 0.0114996928375307)

(-15.647967963898166, 0.0599593189797558)

(50.245178691494786, 0.0203023709567303)

(-6.1441135130178655, -0.138623930394573)

(-12.492390025578958, -0.0739131230459364)

(-53.388691007263, 0.0183830682189117)

(-94.23727990366179, -0.0104995111118831)

(28.237636459574798, -0.0366891865463047)

(-2.8899696976784344, 0.248976134877405)

(-59.67378032644585, 0.0164793457895915)

(-1077.5653530240243, 0.000927157142018311)

(34.52770194677802, -0.0298128246468963)

(31.38302529799723, 0.0328953023371544)

(9.304944683395044, -0.119546681963348)

(-100.52111506581193, -0.00984968979094353)

(12.479477991102517, 0.0867833198945747)

(21.94343715678808, -0.0476933188520339)

(75.38478088574516, 0.0134423955413013)

(-78.52724269495707, 0.0125733134820883)

(94.23705466939735, 0.010724732692878)

(81.66901329465364, 0.0123953812433342)

(-72.242978694986, 0.0136519134817116)

(-122.51401741983913, 0.00809598171844709)

(25.091256207905772, 0.0414731225016059)

(69.1003552230555, 0.0146826283229769)

(43.95902335679378, 0.0232716924030311)

(-50.24597120461141, -0.0195100148956696)

(78.5269183093816, -0.0128976727485698)

(40.81559398815024, -0.0251078697468112)

(59.673218517069586, -0.0170410762454831)

(65.9580523911179, -0.0153927263543733)

(72.24259540785, -0.0140351638863266)

(37.67184972638089, 0.0272587398500595)

(18.793414411369753, 0.0561120230339157)

(-62.81618434806106, -0.0156680826074814)

(-43.960058853137774, -0.022236464203186)

(-97.37920786188297, 0.0101642243790071)

(-28.240147652627645, 0.0341795711715136)

(-87.9533529268738, -0.0112411368826843)

(-75.38513288119637, -0.0130904310684593)

(-546.6352956938762, -0.00182602973305305)

(-18.79909143578314, -0.0504430691319447)

(62.815677356778, 0.0161750096209984)

(-84.81134870414938, 0.0116526790492257)

(84.81107061511238, -0.0119307487512748)

(6.089161203099427, 0.192809042427521)

(53.387989084075315, -0.0190848682073296)

(-47.10310411861372, 0.0207841885412821)

(47.10220226696507, -0.0216858368023364)

(-21.947598583794207, 0.0435362264748061)

(-65.95851221463039, 0.0149329557083856)

(-9.328257063239425, 0.0963710979823201)

(-34.5293808983144, 0.0281345781753277)

(-81.66931319634023, -0.0120955020439642)

(-40.81679521724192, 0.023907001519389)

(97.37899692901101, -0.0103751461271118)

(-31.385057906339963, -0.0308637274812354)

(-56.53128766851124, -0.0173792211238612)

(-91.09532906682657, 0.0108576739325778)


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=−25.0944376288815x_{1} = -25.0944376288815
x2=2.57625015820118x_{2} = 2.57625015820118
x3=−37.673259943911x_{3} = -37.673259943911
x4=191.631906205502x_{4} = 191.631906205502
x5=15.6397620877646x_{5} = 15.6397620877646
x6=−69.1007741687956x_{6} = -69.1007741687956
x7=91.0950880256329x_{7} = 91.0950880256329
x8=−6.14411351301787x_{8} = -6.14411351301787
x9=−12.492390025579x_{9} = -12.492390025579
x10=−94.2372799036618x_{10} = -94.2372799036618
x11=28.2376364595748x_{11} = 28.2376364595748
x12=34.527701946778x_{12} = 34.527701946778
x13=9.30494468339504x_{13} = 9.30494468339504
x14=−100.521115065812x_{14} = -100.521115065812
x15=21.9434371567881x_{15} = 21.9434371567881
x16=−50.2459712046114x_{16} = -50.2459712046114
x17=78.5269183093816x_{17} = 78.5269183093816
x18=40.8155939881502x_{18} = 40.8155939881502
x19=59.6732185170696x_{19} = 59.6732185170696
x20=65.9580523911179x_{20} = 65.9580523911179
x21=72.24259540785x_{21} = 72.24259540785
x22=−62.8161843480611x_{22} = -62.8161843480611
x23=−43.9600588531378x_{23} = -43.9600588531378
x24=−87.9533529268738x_{24} = -87.9533529268738
x25=−75.3851328811964x_{25} = -75.3851328811964
x26=−546.635295693876x_{26} = -546.635295693876
x27=−18.7990914357831x_{27} = -18.7990914357831
x28=84.8110706151124x_{28} = 84.8110706151124
x29=53.3879890840753x_{29} = 53.3879890840753
x30=47.1022022669651x_{30} = 47.1022022669651
x31=−81.6693131963402x_{31} = -81.6693131963402
x32=97.378996929011x_{32} = 97.378996929011
x33=−31.38505790634x_{33} = -31.38505790634
x34=−56.5312876685112x_{34} = -56.5312876685112
Maxima of the function at points:
x34=100.520917114109x_{34} = 100.520917114109
x34=188.490225654589x_{34} = 188.490225654589
x34=56.5306616416093x_{34} = 56.5306616416093
x34=87.9530943542027x_{34} = 87.9530943542027
x34=−15.6479679638982x_{34} = -15.6479679638982
x34=50.2451786914948x_{34} = 50.2451786914948
x34=−53.388691007263x_{34} = -53.388691007263
x34=−2.88996969767843x_{34} = -2.88996969767843
x34=−59.6737803264459x_{34} = -59.6737803264459
x34=−1077.56535302402x_{34} = -1077.56535302402
x34=31.3830252979972x_{34} = 31.3830252979972
x34=12.4794779911025x_{34} = 12.4794779911025
x34=75.3847808857452x_{34} = 75.3847808857452
x34=−78.5272426949571x_{34} = -78.5272426949571
x34=94.2370546693974x_{34} = 94.2370546693974
x34=81.6690132946536x_{34} = 81.6690132946536
x34=−72.242978694986x_{34} = -72.242978694986
x34=−122.514017419839x_{34} = -122.514017419839
x34=25.0912562079058x_{34} = 25.0912562079058
x34=69.1003552230555x_{34} = 69.1003552230555
x34=43.9590233567938x_{34} = 43.9590233567938
x34=37.6718497263809x_{34} = 37.6718497263809
x34=18.7934144113698x_{34} = 18.7934144113698
x34=−97.379207861883x_{34} = -97.379207861883
x34=−28.2401476526276x_{34} = -28.2401476526276
x34=62.815677356778x_{34} = 62.815677356778
x34=−84.8113487041494x_{34} = -84.8113487041494
x34=6.08916120309943x_{34} = 6.08916120309943
x34=−47.1031041186137x_{34} = -47.1031041186137
x34=−21.9475985837942x_{34} = -21.9475985837942
x34=−65.9585122146304x_{34} = -65.9585122146304
x34=−9.32825706323943x_{34} = -9.32825706323943
x34=−34.5293808983144x_{34} = -34.5293808983144
x34=−40.8167952172419x_{34} = -40.8167952172419
x34=−91.0953290668266x_{34} = -91.0953290668266
Decreasing at intervals
[191.631906205502,∞)\left[191.631906205502, \infty\right)
Increasing at intervals
(−∞,−546.635295693876]\left(-\infty, -546.635295693876\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=−237.181848301852x_{1} = -237.181848301852
x2=95.7974767616183x_{2} = 95.7974767616183
x3=4.00507341668955x_{3} = 4.00507341668955
x4=−73.80068645168x_{4} = -73.80068645168
x5=287.448745694871x_{5} = 287.448745694871
x6=48.6527034788051x_{6} = 48.6527034788051
x7=7.54372449628009x_{7} = 7.54372449628009
x8=−32.9277399444348x_{8} = -32.9277399444348
x9=567.053940733425x_{9} = 567.053940733425
x10=80.0853208283276x_{10} = 80.0853208283276
x11=67.5141687409854x_{11} = 67.5141687409854
x12=−20.3264348242219x_{12} = -20.3264348242219
x13=−92.6556268279389x_{13} = -92.6556268279389
x14=−23.4801553706306x_{14} = -23.4801553706306
x15=−29.7801075137773x_{15} = -29.7801075137773
x16=−14.0034717913284x_{16} = -14.0034717913284
x17=73.7999513585394x_{17} = 73.7999513585394
x18=−58.0856084395179x_{18} = -58.0856084395179
x19=23.4728313498836x_{19} = 23.4728313498836
x20=−26.6310922236611x_{20} = -26.6310922236611
x21=58.0844210975337x_{21} = 58.0844210975337
x22=−4.32863617605124x_{22} = -4.32863617605124
x23=83.2278802717944x_{23} = 83.2278802717944
x24=13.9825085391948x_{24} = 13.9825085391948
x25=36.0712578833702x_{25} = 36.0712578833702
x26=105.224163309626x_{26} = 105.224163309626
x27=39.2175523279643x_{27} = 39.2175523279643
x28=89.5127931011103x_{28} = 89.5127931011103
x29=−80.0859449790141x_{29} = -80.0859449790141
x30=76.9426813176863x_{30} = 76.9426813176863
x31=−95.797912862081x_{31} = -95.797912862081
x32=−76.9433575383977x_{32} = -76.9433575383977
x33=−61.2289118119026x_{33} = -61.2289118119026
x34=−168.063376807947x_{34} = -168.063376807947
x35=−70.657920700132x_{35} = -70.657920700132
x36=−48.654396838104x_{36} = -48.654396838104
x37=−10.825651157762x_{37} = -10.825651157762
x38=−42.3653647291314x_{38} = -42.3653647291314
x39=−39.22016138731x_{39} = -39.22016138731
x40=98.9397464504172x_{40} = 98.9397464504172
x41=−51.7983897861238x_{41} = -51.7983897861238
x42=10.7898786754269x_{42} = 10.7898786754269
x43=−7.61991323310644x_{43} = -7.61991323310644
x44=−54.9421125829153x_{44} = -54.9421125829153
x45=17.1546413657741x_{45} = 17.1546413657741
x46=20.3166301288662x_{46} = 20.3166301288662
x47=26.6254109350763x_{47} = 26.6254109350763
x48=−89.5132926274963x_{48} = -89.5132926274963
x49=−45.5100787997204x_{49} = -45.5100787997204
x50=61.2278434114583x_{50} = 61.2278434114583
x51=51.7968961320869x_{51} = 51.7968961320869
x52=45.5081427660817x_{52} = 45.5081427660817
x53=−64.3720505127272x_{53} = -64.3720505127272
x54=86.3703684986956x_{54} = 86.3703684986956
x55=−86.3709050594723x_{55} = -86.3709050594723
x56=−36.0743437126941x_{56} = -36.0743437126941
x57=−17.1684571899007x_{57} = -17.1684571899007
x58=42.3631297553676x_{58} = 42.3631297553676
x59=32.9240332040206x_{59} = 32.9240332040206
x60=54.9407852505616x_{60} = 54.9407852505616
x61=70.6571186927646x_{61} = 70.6571186927646
x62=64.3710840254309x_{62} = 64.3710840254309
x63=−105.224524739209x_{63} = -105.224524739209
x64=−67.5150472396589x_{64} = -67.5150472396589
x65=−98.9401552763972x_{65} = -98.9401552763972
x66=−83.228458145445x_{66} = -83.228458145445
x67=92.6551606286758x_{67} = 92.6551606286758
x68=29.7755709323142x_{68} = 29.7755709323142
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
[567.053940733425,∞)\left[567.053940733425, \infty\right)
Convex at the intervals
(−∞,−95.797912862081]\left(-\infty, -95.797912862081\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)−x−1\frac{\cos{\left(x \right)}}{x - 1} = \frac{\cos{\left(x \right)}}{- x - 1}
- No
cos⁡(x)x−1=−cos⁡(x)−x−1\frac{\cos{\left(x \right)}}{x - 1} = - \frac{\cos{\left(x \right)}}{- x - 1}
- No
so, the function
not is
neither even, nor odd
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