# Partial derivative and limits

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Hello . I have problems with two exercises .
1.$$\lim_{t \to 0 } \frac{2v_1-t^2v_2^2}{|t| \sqrt{v_1^2+v_2^2} }$$
Here, I have to write when this limit will be exist.
2.$$\lim_{(h,k) \to (0,0) } \frac{2hk}{(|h|^a+|k|^a) \cdot \sqrt{h^2+k^2} }$$
Here, I have to write for which $$a \in \mathbb{R}_+$$ this limit will equal to zero.
I don't have ideas how to do it.

stevendaryl
Staff Emeritus
Hello . I have problems with two exercises .
1.$$\lim_{t \to 0 } \frac{2v_1-t^2v_2^2}{|t| \sqrt{v_1^2+v_2^2} }$$
Here, I have to write when this limit will be exist.

Well, in a fraction, as the denominator approaches zero, then the fraction becomes undefined, unless the numerator also approaches zero. So under what circumstances does the numerator go to zero as $t \rightarrow 0$?

pawlo392
Yes. Now I know. When $$v_1=0$$ this limit will equal to zero.

Mark44
Mentor
Yes. Now I know. When $$v_1=0$$ this limit will equal to zero.
But the limit is as t approaches 0. As far as the limit process is concerned, ##v_1## is just some constant. You can't arbitrarily say it's zero.

stevendaryl
Staff Emeritus
But the limit is as t approaches 0. As far as the limit process is concerned, ##v_1## is just some constant. You can't arbitrarily say it's zero.

The question was when (in what circumstances) the limit exists. When $v_1 = 0$ is a possible circumstance.

Ray Vickson
Homework Helper
Dearly Missed
Hello . I have problems with two exercises .
1.$$\lim_{t \to 0 } \frac{2v_1-t^2v_2^2}{|t| \sqrt{v_1^2+v_2^2} }$$
Here, I have to write when this limit will be exist.
2.$$\lim_{(h,k) \to (0,0) } \frac{2hk}{(|h|^a+|k|^a) \cdot \sqrt{h^2+k^2} }$$
Here, I have to write for which $$a \in \mathbb{R}_+$$ this limit will equal to zero.
I don't have ideas how to do it.

For the second one, I would use polar coordinates ##h = r \cos \theta, k = r \sin \theta##, so that we are taking the limit as ##r \to 0##.

pawlo392