Answered: "The problem is:lim(x,y,z)→(0,0,0)xy+2yz+3xzx2+4y2+9z2"

George Michael

Answered question

2022-04-10

The problem is:

lim_{(x, y, z) \to (0, 0, 0)} \frac{x y + 2 y z + 3 x z}{x^{2} + 4 y^{2} + 9 z^{2}}

ditumpasz9xj

Beginner2022-04-11Added 8 answers

Let's take the limit

z \to 0

first, getting

lim_{x, y \to 0} \frac{x y}{x^{2} + 4 y^{2}}

Now consider what happens if you take the limit along

y = x

lim_{x, y \to 0} \frac{x y}{x^{2} + 4 y^{2}} = lim_{x \to 0} \frac{x^{2}}{5 x^{2}} = \frac{1}{5}

and along

y = 2 x

lim_{x, y \to 0} \frac{x y}{x^{2} + 4 y^{2}} = lim_{x \to 0} \frac{2 x^{2}}{17 x^{2}} = \frac{2}{17}

That's enough to tell you that the limit depends on direction.

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