What Conditions Determine the Existence of These Mathematical Limits?

[pawlo392]

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]

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]

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]

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]

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 $r \to 0$.