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Plotting a function graph/ sin(1/x)

Graphing y = sin(1/x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /1\
f(x) = sin|-|
          \x/
f(x)=sin⁡(1x)f{\left(x \right)} = \sin{\left(\frac{1}{x} \right)} 
f = sin(1/x)
The graph of the function
02468-8-6-4-2-10102-2
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:
sin⁡(1x)=0\sin{\left(\frac{1}{x} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1πx_{1} = \frac{1}{\pi}
Numerical solution
x1=0.318309886183791x_{1} = 0.318309886183791
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(1/x).
sin⁡(10)\sin{\left(\frac{1}{0} \right)}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
−cos⁡(1x)x2=0- \frac{\cos{\left(\frac{1}{x} \right)}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=23πx_{1} = \frac{2}{3 \pi}
x2=2πx_{2} = \frac{2}{\pi}
The values of the extrema at the points:
  2       
(----, -1)
 3*pi     

 2     
(--, 1)
 pi


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=23πx_{1} = \frac{2}{3 \pi}
Maxima of the function at points:
x1=2πx_{1} = \frac{2}{\pi}
Decreasing at intervals
[23π,2π]\left[\frac{2}{3 \pi}, \frac{2}{\pi}\right]
Increasing at intervals
(−∞,23π]∪[2π,∞)\left(-\infty, \frac{2}{3 \pi}\right] \cup \left[\frac{2}{\pi}, \infty\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
2cos⁡(1x)−sin⁡(1x)xx3=0\frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=33124.297012115x_{1} = 33124.297012115
x2=39904.771377312x_{2} = 39904.771377312
x3=14479.7313150492x_{3} = 14479.7313150492
x4=21259.0191811792x_{4} = 21259.0191811792
x5=−36383.2913419629x_{5} = -36383.2913419629
x6=−27060.3085181287x_{6} = -27060.3085181287
x7=−18585.4480940284x_{7} = -18585.4480940284
x8=28886.5862895034x_{8} = 28886.5862895034
x9=25496.4920696083x_{9} = 25496.4920696083
x10=−32145.5241220507x_{10} = -32145.5241220507
x11=17869.2283625365x_{11} = 17869.2283625365
x12=−31297.9789349291x_{12} = -31297.9789349291
x13=19564.09661214x_{13} = 19564.09661214
x14=15327.0673496263x_{14} = 15327.0673496263
x15=−39773.5440970563x_{15} = -39773.5440970563
x16=38209.640819107x_{16} = 38209.640819107
x17=−14348.5424650725x_{17} = -14348.5424650725
x18=−38925.9782260908x_{18} = -38925.9782260908
x19=−37230.8517263672x_{19} = -37230.8517263672
x20=−25365.273191964x_{20} = -25365.273191964
x21=20411.5519538498x_{21} = 20411.5519538498
x22=−12653.9899632038x_{22} = -12653.9899632038
x23=−19432.887686874x_{23} = -19432.887686874
x24=−17738.0242579883x_{24} = -17738.0242579883
x25=11937.9523841008x_{25} = 11937.9523841008
x26=−40621.1115695252x_{26} = -40621.1115695252
x27=41599.9082873259x_{27} = 41599.9082873259
x28=−11806.7845665482x_{28} = -11806.7845665482
x29=23801.4794641224x_{29} = 23801.4794641224
x30=16174.4315462694x_{30} = 16174.4315462694
x31=36514.5174977468x_{31} = 36514.5174977468
x32=32276.7483423665x_{32} = 32276.7483423665
x33=−33840.6233739026x_{33} = -33840.6233739026
x34=−26212.7880538116x_{34} = -26212.7880538116
x35=−32993.0723444395x_{35} = -32993.0723444395
x36=42447.4788863631x_{36} = 42447.4788863631
x37=37362.078192229x_{37} = 37362.078192229
x38=24648.982396646x_{38} = 24648.982396646
x39=−24517.764514177x_{39} = -24517.764514177
x40=−13501.2455714192x_{40} = -13501.2455714192
x41=−41468.6805452608x_{41} = -41468.6805452608
x42=−21975.2827488203x_{42} = -21975.2827488203
x43=33971.8484557919x_{43} = 33971.8484557919
x44=−20280.3410578956x_{44} = -20280.3410578956
x45=−21127.8065465298x_{45} = -21127.8065465298
x46=−15195.8737165349x_{46} = -15195.8737165349
x47=−42316.2509339008x_{47} = -42316.2509339008
x48=−34688.1770045647x_{48} = -34688.1770045647
x49=−16890.6185554691x_{49} = -16890.6185554691
x50=18716.6547734102x_{50} = 18716.6547734102
x51=−30450.4370366936x_{51} = -30450.4370366936
x52=13632.4287111505x_{52} = 13632.4287111505
x53=−38078.4140636161x_{53} = -38078.4140636161
x54=28039.0554132717x_{54} = 28039.0554132717
x55=17021.8196886792x_{55} = 17021.8196886792
x56=22953.9840195429x_{56} = 22953.9840195429
x57=35666.9588734903x_{57} = 35666.9588734903
x58=30581.6602478055x_{58} = 30581.6602478055
x59=27191.5291133465x_{59} = 27191.5291133465
x60=−22822.7684696829x_{60} = -22822.7684696829
x61=29734.1213504079x_{61} = 29734.1213504079
x62=39057.2052525216x_{62} = 39057.2052525216
x63=−23670.2626854463x_{63} = -23670.2626854463
x64=22106.496925216x_{64} = 22106.496925216
x65=−29602.8987100326x_{65} = -29602.8987100326
x66=−28755.3642709925x_{66} = -28755.3642709925
x67=−16043.2338667424x_{67} = -16043.2338667424
x68=31429.2026711056x_{68} = 31429.2026711056
x69=26344.0078318137x_{69} = 26344.0078318137
x70=−35535.7330502043x_{70} = -35535.7330502043
x71=12785.166209279x_{71} = 12785.166209279
x72=40752.3390879025x_{72} = 40752.3390879025
x73=34819.4024707177x_{73} = 34819.4024707177
x74=−27907.8340740483x_{74} = -27907.8340740483
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

True

True

- 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:
Have no bends at the whole real axis
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→−∞sin⁡(1x)=0\lim_{x \to -\infty} \sin{\left(\frac{1}{x} \right)} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞sin⁡(1x)=0\lim_{x \to \infty} \sin{\left(\frac{1}{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 sin(1/x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(1x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(1x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x}\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:
sin⁡(1x)=−sin⁡(1x)\sin{\left(\frac{1}{x} \right)} = - \sin{\left(\frac{1}{x} \right)}
- No
sin⁡(1x)=sin⁡(1x)\sin{\left(\frac{1}{x} \right)} = \sin{\left(\frac{1}{x} \right)}
- Yes
so, the function
is
odd
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