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

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(x - 1)
f(x) = ----------
         x - 1
f(x)=cos⁡(x−1)x−1f{\left(x \right)} = \frac{\cos{\left(x - 1 \right)}}{x - 1} 
f = cos(x - 1)/(x - 1)
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:
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−1)x−1=0\frac{\cos{\left(x - 1 \right)}}{x - 1} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1+π2x_{1} = 1 + \frac{\pi}{2}
x2=1+3π2x_{2} = 1 + \frac{3 \pi}{2}
Numerical solution
x1=−44.553093477052x_{1} = -44.553093477052
x2=−9.99557428756428x_{2} = -9.99557428756428
x3=71.6858347057703x_{3} = 71.6858347057703
x4=−53.9778714378214x_{4} = -53.9778714378214
x5=−69.6858347057703x_{5} = -69.6858347057703
x6=125.092909816797x_{6} = 125.092909816797
x7=30.845130209103x_{7} = 30.845130209103
x8=−50.8362787842316x_{8} = -50.8362787842316
x9=−41.4115008234622x_{9} = -41.4115008234622
x10=−25.7035375555132x_{10} = -25.7035375555132
x11=−38.2699081698724x_{11} = -38.2699081698724
x12=−82.2522053201295x_{12} = -82.2522053201295
x13=357.570766182442x_{13} = 357.570766182442
x14=96.8185759344887x_{14} = 96.8185759344887
x15=33.9867228626928x_{15} = 33.9867228626928
x16=65.4026493985908x_{16} = 65.4026493985908
x17=−72.8274273593601x_{17} = -72.8274273593601
x18=−6.85398163397448x_{18} = -6.85398163397448
x19=−0.570796326794897x_{19} = -0.570796326794897
x20=74.8274273593601x_{20} = 74.8274273593601
x21=21.4203522483337x_{21} = 21.4203522483337
x22=134.517687777566x_{22} = 134.517687777566
x23=59.1194640914112x_{23} = 59.1194640914112
x24=−28.845130209103x_{24} = -28.845130209103
x25=109.384946548848x_{25} = 109.384946548848
x26=18.2787595947439x_{26} = 18.2787595947439
x27=−75.9690200129499x_{27} = -75.9690200129499
x28=93.6769832808989x_{28} = 93.6769832808989
x29=−66.5442420521806x_{29} = -66.5442420521806
x30=−19.4203522483337x_{30} = -19.4203522483337
x31=99.9601685880785x_{31} = 99.9601685880785
x32=−47.6946861306418x_{32} = -47.6946861306418
x33=84.2522053201295x_{33} = 84.2522053201295
x34=55.9778714378214x_{34} = 55.9778714378214
x35=−88.5353906273091x_{35} = -88.5353906273091
x36=5.71238898038469x_{36} = 5.71238898038469
x37=−214.199096770901x_{37} = -214.199096770901
x38=−60.261056745001x_{38} = -60.261056745001
x39=81.1106126665397x_{39} = 81.1106126665397
x40=52.8362787842316x_{40} = 52.8362787842316
x41=90.5353906273091x_{41} = 90.5353906273091
x42=−22.5619449019235x_{42} = -22.5619449019235
x43=−94.8185759344887x_{43} = -94.8185759344887
x44=62.261056745001x_{44} = 62.261056745001
x45=49.6946861306418x_{45} = 49.6946861306418
x46=15.1371669411541x_{46} = 15.1371669411541
x47=40.2699081698724x_{47} = 40.2699081698724
x48=27.7035375555132x_{48} = 27.7035375555132
x49=46.553093477052x_{49} = 46.553093477052
x50=43.4115008234622x_{50} = 43.4115008234622
x51=68.5442420521806x_{51} = 68.5442420521806
x52=−35.1283155162826x_{52} = -35.1283155162826
x53=−97.9601685880785x_{53} = -97.9601685880785
x54=−91.6769832808989x_{54} = -91.6769832808989
x55=2.5707963267949x_{55} = 2.5707963267949
x56=11.9955742875643x_{56} = 11.9955742875643
x57=87.3937979737193x_{57} = 87.3937979737193
x58=−101.101761241668x_{58} = -101.101761241668
x59=−85.3937979737193x_{59} = -85.3937979737193
x60=8.85398163397448x_{60} = 8.85398163397448
x61=−16.2787595947439x_{61} = -16.2787595947439
x62=103.101761241668x_{62} = 103.101761241668
x63=77.9690200129499x_{63} = 77.9690200129499
x64=−63.4026493985908x_{64} = -63.4026493985908
x65=24.5619449019235x_{65} = 24.5619449019235
x66=37.1283155162826x_{66} = 37.1283155162826
x67=−3.71238898038469x_{67} = -3.71238898038469
x68=−57.1194640914112x_{68} = -57.1194640914112
x69=−13.1371669411541x_{69} = -13.1371669411541
x70=−31.9867228626928x_{70} = -31.9867228626928
x71=−79.1106126665397x_{71} = -79.1106126665397
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x - 1)/(x - 1).
cos⁡(−1)−1\frac{\cos{\left(-1 \right)}}{-1}
The result:
f(0)=−cos⁡(1)f{\left(0 \right)} = - \cos{\left(1 \right)}
The point:
(0, -cos(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−1)x−1−cos⁡(x−1)(x−1)2=0- \frac{\sin{\left(x - 1 \right)}}{x - 1} - \frac{\cos{\left(x - 1 \right)}}{\left(x - 1\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=−27.2389365752603x_{1} = -27.2389365752603
x2=73.2427897046973x_{2} = 73.2427897046973
x3=82.6691650818489x_{3} = 82.6691650818489
x4=−55.5309801938186x_{4} = -55.5309801938186
x5=7.12125046689807x_{5} = 7.12125046689807
x6=−30.3840740178899x_{6} = -30.3840740178899
x7=44.9595528888955x_{7} = 44.9595528888955
x8=85.811211299318x_{8} = 85.811211299318
x9=35.5285657554621x_{9} = 35.5285657554621
x10=−46.1026627703624x_{10} = -46.1026627703624
x11=−99.5210170746866x_{11} = -99.5210170746866
x12=95.2371684817036x_{12} = 95.2371684817036
x13=70.100567727981x_{13} = 70.100567727981
x14=22.945612879981x_{14} = 22.945612879981
x15=−93.2371684817036x_{15} = -93.2371684817036
x16=−49.2455828375744x_{16} = -49.2455828375744
x17=32.3840740178899x_{17} = 32.3840740178899
x18=−39.8162093266346x_{18} = -39.8162093266346
x19=−83.811211299318x_{19} = -83.811211299318
x20=−52.3883466217256x_{20} = -52.3883466217256
x21=−5.12125046689807x_{21} = -5.12125046689807
x22=334.005818339011x_{22} = 334.005818339011
x23=98.3791034786112x_{23} = 98.3791034786112
x24=−1.79838604578389x_{24} = -1.79838604578389
x25=3.79838604578389x_{25} = 3.79838604578389
x26=10.3178664617911x_{26} = 10.3178664617911
x27=26.0929104121121x_{27} = 26.0929104121121
x28=48.1026627703624x_{28} = 48.1026627703624
x29=−33.5285657554621x_{29} = -33.5285657554621
x30=−77.5270825679419x_{30} = -77.5270825679419
x31=60.6735041304405x_{31} = 60.6735041304405
x32=16.644128370333x_{32} = 16.644128370333
x33=−96.3791034786112x_{33} = -96.3791034786112
x34=19.7964043662102x_{34} = 19.7964043662102
x35=−71.2427897046973x_{35} = -71.2427897046973
x36=−130.939312353727x_{36} = -130.939312353727
x37=63.8159348889734x_{37} = 63.8159348889734
x38=38.672573565113x_{38} = 38.672573565113
x39=−14.644128370333x_{39} = -14.644128370333
x40=−42.9595528888955x_{40} = -42.9595528888955
x41=−8.31786646179107x_{41} = -8.31786646179107
x42=−80.6691650818489x_{42} = -80.6691650818489
x43=−61.8159348889734x_{43} = -61.8159348889734
x44=−90.0952098694071x_{44} = -90.0952098694071
x45=57.5309801938186x_{45} = 57.5309801938186
x46=76.3849592185347x_{46} = 76.3849592185347
x47=13.4864543952238x_{47} = 13.4864543952238
x48=88.9532251106725x_{48} = 88.9532251106725
x49=−11.4864543952238x_{49} = -11.4864543952238
x50=29.2389365752603x_{50} = 29.2389365752603
x51=−68.100567727981x_{51} = -68.100567727981
x52=−24.0929104121121x_{52} = -24.0929104121121
x53=79.5270825679419x_{53} = 79.5270825679419
x54=−58.6735041304405x_{54} = -58.6735041304405
x55=54.3883466217256x_{55} = 54.3883466217256
x56=−86.9532251106725x_{56} = -86.9532251106725
x57=51.2455828375744x_{57} = 51.2455828375744
x58=−64.9582857893902x_{58} = -64.9582857893902
x59=41.8162093266346x_{59} = 41.8162093266346
x60=−17.7964043662102x_{60} = -17.7964043662102
x61=92.0952098694071x_{61} = 92.0952098694071
x62=66.9582857893902x_{62} = 66.9582857893902
x63=−74.3849592185347x_{63} = -74.3849592185347
x64=−20.945612879981x_{64} = -20.945612879981
x65=−36.672573565113x_{65} = -36.672573565113
The values of the extrema at the points:
(-27.238936575260272, 0.0353899155541688)

(73.24278970469729, -0.0138408859131547)

(82.66916508184887, 0.0122436055670467)

(-55.53098019381864, -0.0176866485521696)

(7.1212504668980685, 0.161228034325064)

(-30.38407401788986, -0.0318471321112693)

(44.959552888895495, 0.0227423004725314)

(85.81121129931802, -0.0117900744410766)

(35.52856575546206, -0.0289493889114503)

(-46.10266277036235, 0.0212254394164143)

(-99.52101707468658, -0.00994767611536293)

(95.23716848170359, 0.01061092686295)

(70.10056772798097, 0.0144701459746764)

(22.945612879981045, -0.0455199604051285)

(-93.23716848170359, -0.01061092686295)

(-49.24558283757444, -0.0198983065303553)

(32.38407401788986, 0.0318471321112693)

(-39.81620932663458, 0.0244927205346957)

(-83.81121129931802, 0.0117900744410766)

(-52.38834662172563, 0.0187273944640866)

(-5.1212504668980685, -0.161228034325064)

(334.00581833901066, 0.00300293699420144)

(98.3791034786112, -0.0102686022030809)

(-1.7983860457838872, 0.336508416918395)

(3.798386045783887, -0.336508416918395)

(10.317866461791066, -0.106707947715237)

(26.092910412112097, 0.0398202855500511)

(48.10266277036235, -0.0212254394164143)

(-33.52856575546206, 0.0289493889114503)

(-77.52708256794193, 0.0127334276777468)

(60.67350413044053, -0.0167555036571887)

(16.64412837033303, -0.0637915530395936)

(-96.3791034786112, 0.0102686022030809)

(19.796404366210158, 0.0531265325613881)

(-71.24278970469729, 0.0138408859131547)

(-130.9393123537265, -0.00757902448438246)

(63.81593488897342, 0.015917510583426)

(38.67257356511297, 0.0265351630103045)

(-14.644128370333028, 0.0637915530395936)

(-42.959552888895495, -0.0227423004725314)

(-8.317866461791066, 0.106707947715237)

(-80.66916508184887, -0.0122436055670467)

(-61.81593488897342, -0.015917510583426)

(-90.09520986940714, 0.0109768642483425)

(57.53098019381864, 0.0176866485521696)

(76.38495921853475, 0.0132640786518247)

(13.486454395223781, 0.0798311807800032)

(88.95322511067255, 0.0113689449158811)

(-11.486454395223781, -0.0798311807800032)

(29.238936575260272, -0.0353899155541688)

(-68.10056772798097, -0.0144701459746764)

(-24.092910412112097, -0.0398202855500511)

(79.52708256794193, -0.0127334276777468)

(-58.67350413044053, 0.0167555036571887)

(54.38834662172563, -0.0187273944640866)

(-86.95322511067255, -0.0113689449158811)

(51.24558283757444, 0.0198983065303553)

(-64.95828578939016, 0.0151593553168405)

(41.81620932663458, -0.0244927205346957)

(-17.796404366210158, -0.0531265325613881)

(92.09520986940714, -0.0109768642483425)

(66.95828578939016, -0.0151593553168405)

(-74.38495921853475, -0.0132640786518247)

(-20.945612879981045, 0.0455199604051285)

(-36.67257356511297, -0.0265351630103045)


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=73.2427897046973x_{1} = 73.2427897046973
x2=−55.5309801938186x_{2} = -55.5309801938186
x3=−30.3840740178899x_{3} = -30.3840740178899
x4=85.811211299318x_{4} = 85.811211299318
x5=35.5285657554621x_{5} = 35.5285657554621
x6=−99.5210170746866x_{6} = -99.5210170746866
x7=22.945612879981x_{7} = 22.945612879981
x8=−93.2371684817036x_{8} = -93.2371684817036
x9=−49.2455828375744x_{9} = -49.2455828375744
x10=−5.12125046689807x_{10} = -5.12125046689807
x11=98.3791034786112x_{11} = 98.3791034786112
x12=3.79838604578389x_{12} = 3.79838604578389
x13=10.3178664617911x_{13} = 10.3178664617911
x14=48.1026627703624x_{14} = 48.1026627703624
x15=60.6735041304405x_{15} = 60.6735041304405
x16=16.644128370333x_{16} = 16.644128370333
x17=−130.939312353727x_{17} = -130.939312353727
x18=−42.9595528888955x_{18} = -42.9595528888955
x19=−80.6691650818489x_{19} = -80.6691650818489
x20=−61.8159348889734x_{20} = -61.8159348889734
x21=−11.4864543952238x_{21} = -11.4864543952238
x22=29.2389365752603x_{22} = 29.2389365752603
x23=−68.100567727981x_{23} = -68.100567727981
x24=−24.0929104121121x_{24} = -24.0929104121121
x25=79.5270825679419x_{25} = 79.5270825679419
x26=54.3883466217256x_{26} = 54.3883466217256
x27=−86.9532251106725x_{27} = -86.9532251106725
x28=41.8162093266346x_{28} = 41.8162093266346
x29=−17.7964043662102x_{29} = -17.7964043662102
x30=92.0952098694071x_{30} = 92.0952098694071
x31=66.9582857893902x_{31} = 66.9582857893902
x32=−74.3849592185347x_{32} = -74.3849592185347
x33=−36.672573565113x_{33} = -36.672573565113
Maxima of the function at points:
x33=−27.2389365752603x_{33} = -27.2389365752603
x33=82.6691650818489x_{33} = 82.6691650818489
x33=7.12125046689807x_{33} = 7.12125046689807
x33=44.9595528888955x_{33} = 44.9595528888955
x33=−46.1026627703624x_{33} = -46.1026627703624
x33=95.2371684817036x_{33} = 95.2371684817036
x33=70.100567727981x_{33} = 70.100567727981
x33=32.3840740178899x_{33} = 32.3840740178899
x33=−39.8162093266346x_{33} = -39.8162093266346
x33=−83.811211299318x_{33} = -83.811211299318
x33=−52.3883466217256x_{33} = -52.3883466217256
x33=334.005818339011x_{33} = 334.005818339011
x33=−1.79838604578389x_{33} = -1.79838604578389
x33=26.0929104121121x_{33} = 26.0929104121121
x33=−33.5285657554621x_{33} = -33.5285657554621
x33=−77.5270825679419x_{33} = -77.5270825679419
x33=−96.3791034786112x_{33} = -96.3791034786112
x33=19.7964043662102x_{33} = 19.7964043662102
x33=−71.2427897046973x_{33} = -71.2427897046973
x33=63.8159348889734x_{33} = 63.8159348889734
x33=38.672573565113x_{33} = 38.672573565113
x33=−14.644128370333x_{33} = -14.644128370333
x33=−8.31786646179107x_{33} = -8.31786646179107
x33=−90.0952098694071x_{33} = -90.0952098694071
x33=57.5309801938186x_{33} = 57.5309801938186
x33=76.3849592185347x_{33} = 76.3849592185347
x33=13.4864543952238x_{33} = 13.4864543952238
x33=88.9532251106725x_{33} = 88.9532251106725
x33=−58.6735041304405x_{33} = -58.6735041304405
x33=51.2455828375744x_{33} = 51.2455828375744
x33=−64.9582857893902x_{33} = -64.9582857893902
x33=−20.945612879981x_{33} = -20.945612879981
Decreasing at intervals
[98.3791034786112,∞)\left[98.3791034786112, \infty\right)
Increasing at intervals
(−∞,−130.939312353727]\left(-\infty, -130.939312353727\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−1)+2sin⁡(x−1)x−1+2cos⁡(x−1)(x−1)2x−1=0\frac{- \cos{\left(x - 1 \right)} + \frac{2 \sin{\left(x - 1 \right)}}{x - 1} + \frac{2 \cos{\left(x - 1 \right)}}{\left(x - 1\right)^{2}}}{x - 1} = 0
Solve this equation
The roots of this equation
x1=55.9414610202918x_{1} = 55.9414610202918
x2=−97.9399529307048x_{2} = -97.9399529307048
x3=99.9399529307048x_{3} = 99.9399529307048
x4=−44.5091321154553x_{4} = -44.5091321154553
x5=−72.8003238908837x_{5} = -72.8003238908837
x6=84.2281726832512x_{6} = 84.2281726832512
x7=74.8003238908837x_{7} = 74.8003238908837
x8=−60.2283863503723x_{8} = -60.2283863503723
x9=27.6283591640252x_{9} = 27.6283591640252
x10=−94.7976970894915x_{10} = -94.7976970894915
x11=−88.5130456566371x_{11} = -88.5130456566371
x12=14.9937625671267x_{12} = 14.9937625671267
x13=−91.655396245836x_{13} = -91.655396245836
x14=21.3217772482235x_{14} = 21.3217772482235
x15=−63.3715747870554x_{15} = -63.3715747870554
x16=−19.3217772482235x_{16} = -19.3217772482235
x17=−31.9259431758392x_{17} = -31.9259431758392
x18=87.370639887736x_{18} = 87.370639887736
x19=−22.4766510546492x_{19} = -22.4766510546492
x20=81.0856368040887x_{20} = 81.0856368040887
x21=178.488716367408x_{21} = 178.488716367408
x22=65.3715747870554x_{22} = 65.3715747870554
x23=−75.9430238267933x_{23} = -75.9430238267933
x24=−16.1619600917303x_{24} = -16.1619600917303
x25=11.8095072981602x_{25} = 11.8095072981602
x26=93.655396245836x_{26} = 93.655396245836
x27=49.6535676048409x_{27} = 49.6535676048409
x28=−101.082167928013x_{28} = -101.082167928013
x29=71.6575253785884x_{29} = 71.6575253785884
x30=90.5130456566371x_{30} = 90.5130456566371
x31=−66.5146145048817x_{31} = -66.5146145048817
x32=−50.7976574095537x_{32} = -50.7976574095537
x33=−28.7779159141436x_{33} = -28.7779159141436
x34=40.218890250481x_{34} = 40.218890250481
x35=−79.0856368040887x_{35} = -79.0856368040887
x36=−68845.4321780434x_{36} = -68845.4321780434
x37=−85.370639887736x_{37} = -85.370639887736
x38=37.0728437679879x_{38} = 37.0728437679879
x39=43.3642737086586x_{39} = 43.3642737086586
x40=216.18980251639x_{40} = 216.18980251639
x41=−41.3642737086586x_{41} = -41.3642737086586
x42=−157.637821338313x_{42} = -157.637821338313
x43=62.2283863503723x_{43} = 62.2283863503723
x44=96.7976970894915x_{44} = 96.7976970894915
x45=77.9430238267933x_{45} = 77.9430238267933
x46=−69.6575253785884x_{46} = -69.6575253785884
x47=68.5146145048817x_{47} = 68.5146145048817
x48=52.7976574095537x_{48} = 52.7976574095537
x49=253.890299964068x_{49} = 253.890299964068
x50=−25.6283591640252x_{50} = -25.6283591640252
x51=18.1619600917303x_{51} = 18.1619600917303
x52=−12.9937625671267x_{52} = -12.9937625671267
x53=24.4766510546492x_{53} = 24.4766510546492
x54=−53.9414610202918x_{54} = -53.9414610202918
x55=−6.5873993379941x_{55} = -6.5873993379941
x56=−38.218890250481x_{56} = -38.218890250481
x57=59.085025007445x_{57} = 59.085025007445
x58=−9.80950729816022x_{58} = -9.80950729816022
x59=−82.2281726832512x_{59} = -82.2281726832512
x60=−57.085025007445x_{60} = -57.085025007445
x61=−47.6535676048409x_{61} = -47.6535676048409
x62=−35.0728437679879x_{62} = -35.0728437679879
x63=30.7779159141436x_{63} = 30.7779159141436
x64=5.2222763997912x_{64} = 5.2222763997912
x65=33.9259431758392x_{65} = 33.9259431758392
x66=−3.2222763997912x_{66} = -3.2222763997912
x67=8.5873993379941x_{67} = 8.5873993379941
x68=46.5091321154553x_{68} = 46.5091321154553
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−1)+2sin⁡(x−1)x−1+2cos⁡(x−1)(x−1)2x−1)=−∞\lim_{x \to 1^-}\left(\frac{- \cos{\left(x - 1 \right)} + \frac{2 \sin{\left(x - 1 \right)}}{x - 1} + \frac{2 \cos{\left(x - 1 \right)}}{\left(x - 1\right)^{2}}}{x - 1}\right) = -\infty
lim⁡x→1+(−cos⁡(x−1)+2sin⁡(x−1)x−1+2cos⁡(x−1)(x−1)2x−1)=∞\lim_{x \to 1^+}\left(\frac{- \cos{\left(x - 1 \right)} + \frac{2 \sin{\left(x - 1 \right)}}{x - 1} + \frac{2 \cos{\left(x - 1 \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
[253.890299964068,∞)\left[253.890299964068, \infty\right)
Convex at the intervals
(−∞,−68845.4321780434]\left(-\infty, -68845.4321780434\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−1)x−1)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x - 1 \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−1)x−1)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x - 1 \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 - 1)/(x - 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x−1)x(x−1))=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x - 1 \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−1)x(x−1))=0\lim_{x \to \infty}\left(\frac{\cos{\left(x - 1 \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−1)x−1=cos⁡(x+1)−x−1\frac{\cos{\left(x - 1 \right)}}{x - 1} = \frac{\cos{\left(x + 1 \right)}}{- x - 1}
- No
cos⁡(x−1)x−1=−cos⁡(x+1)−x−1\frac{\cos{\left(x - 1 \right)}}{x - 1} = - \frac{\cos{\left(x + 1 \right)}}{- x - 1}
- No
so, the function
not is
neither even, nor odd
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