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Given $f(x,y)=1-x^2-y^2$, find the directional derivative at the point $(x_0,y_0)$ in the direction of the unit vector $\vec u$.

I am experiencing some strange behavior with the Limit function. I quit the kernel and re-evaluate the notebook, but here is what happens.

Clear[f, x0, y0, a, b, u];
f[x_, y_] = 1 - x^2 - y^2;
a = Cos[Pi/3];
b = Sin[Pi/3];
u = {a, b};
{x0, y0} = .2 u;

Then, the limit definition is: $$D_uf(x_0,y_0)=\lim_{h\to 0}\frac{f(x_0+h a, y_0+h b)-f(x_0,y_0)}{h}$$ However, I am getting a strange answer.

Limit[(f[x0 + h a, y0 + h b] - f[x0, y0])/h, h -> 0]

Output:

\[Infinity]

Now, consider:

Table[(f[x0 + h a, y0 + h b] - f[x0, y0])/
  h, {h, {1., .1, .01, .001, 0.0001}}]

Output:

{-1.4, -0.5, -0.41, -0.401, -0.4001}

Another definition is $D_uf(x_0,y_0)=\nabla f(x_0,y_0)\cdot \vec u$.

Grad[f[x, y], {x, y}].u;
% /. {x -> x0, y -> y0}

Output:

-0.4

See? The correct answer is $-0.4$, but the limit definition is not working for some reason? I hope this isn't a typo or some silly thing.

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The problem is that you're mixing exact and machine numbers in the definition of the function f. The machine numbers create a small nonzero constant term in the numerator of the Limit, which is the cause of the infinite result as you divide by h and take h -> 0.

The fix is to use {x0,y0}= u/5 instead of 0.2.

However, if you do need to work with machine numbers, you could do this:

Needs["NumericalCalculus`"]

{x0,y0}= .2u;

NLimit[(f[x0 + h a, y0 + h b] - f[x0, y0])/h, h -> 0]

(* ==> -0.4 *)

Numerical limits as done in NLimit account for the presence of the kind of roundoff errors that you're seeing.

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    $\begingroup$ Also, you should use SetDelayed in your function definition: f[x_,y_] := 1 - x^2 - y^2. $\endgroup$ – Marius Ladegård Meyer Jul 21 '15 at 5:53
  • $\begingroup$ @MariusLadegårdMeyer That's true, unless x and y are Cleared beforehand. $\endgroup$ – Jens Jul 21 '15 at 5:54
  • $\begingroup$ @Jens. Thanks for the quick answer. I performed each of your suggestions and both work, but this is a bizarre event. I have never experienced this before. This probably will be rare with the students, but when it happens, oh my goodness. Tough one to explain. $\endgroup$ – David Jul 21 '15 at 6:13
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    $\begingroup$ @MariusLadegårdMeyer. I am currently writing on notebook introducing the directional derivative for my students in the fall. I think this event, though bizarre, is actually (as you say) a fortunate event, because this example is going into the notebook with an explanation to the students on how to handle the situation. Thanks for all the help. $\endgroup$ – David Jul 21 '15 at 6:28
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    $\begingroup$ Looking at the result of Simplify[(f[x0 + h a, y0 + h b] - f[x0, y0])/h] should be revealing. $\endgroup$ – george2079 Jul 21 '15 at 14:28

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