I know that the tangent at 0 of x Log [Abs [x]] is -Infinity. But this is not clearly visible on the curve at the scale x = -1 to x = 1.

Indeed, the convergence of the slope towards -Infinity is very slow.

I would really like to highlight this behavior thanks to an astute PlotRange coupled with a Manipulate which makes it possible to observe that the slope becomes infinite when looking at scales always.

I tried to do this:

 Show[Plot[x *Log[Abs[x]], {x, (-10)^-k, 10^-k}, PlotRange -> All], 
  Plot[x - 1, {x, (-10)^-k , 10^-k}], PlotRange -> All], {k, -5, 5}]

but it returns an error message.

  • $\begingroup$ Do you want to plot the function f=x*Log[Abs[x]] or its derivative / slope? I dont see the derivative in your plot command. $\endgroup$ Nov 12 '16 at 23:46
  • $\begingroup$ I want to study f=x*Log[Abs[x]] In the vicinity of 0 $\endgroup$
    – Robin
    Nov 12 '16 at 23:49
  • $\begingroup$ Possible duplicates: (18090), (119514). Related: (19737), (a79760) $\endgroup$
    – Michael E2
    Dec 13 '16 at 14:33

I think you want -10^-k rather than (-10)^-k:

 Show[Plot[x*Log[Abs[x]], {x, -10^-k, 10^-k}, PlotRange -> All], 
  Plot[x - 1, {x, -10^-k, 10^-k}], PlotRange -> All],
 {k, -5, 5}]

Tangent plot

But it might be better if you had legends:

Manipulate[Plot[{x*Log[Abs[x]], x - 1}, {x, -10^-k, 10^-k},
  PlotRange -> All,
  PlotLegends -> {"x*Log[Abs[x]]", "x-1"}],
 {k, -5, 5}]

Plot with legends

However, to get the "feeling" that the slope is $-\infty$ at $x=0$, you want both axes to be on the same scale:

Manipulate[Plot[x*Log[Abs[x]], {x, -10^-k, 10^-k}, AspectRatio -> 1,
  PlotRange -> {{-10^-k, 10^-k}, {-10^-k, 10^-k}}],
 {k, 0, 20}]

Even better display of slope at zero

  • $\begingroup$ Thannnk you it'sperfect, I have been searching for a long time, Thanks a lot.Moreover, I am not sure, but I see that the slope becomes infinite when one looks at scales always smaller. Is ist a good impression ? $\endgroup$
    – Robin
    Nov 12 '16 at 23:54
  • $\begingroup$ That's a good impression but that doesn't become as obvious if the axes are on different scales. I've edited my answer to make that more obvious. $\endgroup$
    – JimB
    Nov 13 '16 at 0:16
f[x_] := Piecewise[{{x Log[-x], x < 0}, {x Log[x], x > 0}}]
der[p_] := D[f[u], u] /. u -> p
fun[p_] := 
 Plot[Evaluate@{f[x], D[f[x], x]}, {x, -2, 2}, 
  Epilog -> {Point[{p, f[p]}], 
    Arrow[{{p, f[p]}, {p, f[p]} + Normalize[{1, der[p]}]}]}]
ListAnimate[Table[fun[j], {j, -2, 2, 0.12}]]

enter image description here


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