zdebe5l8

2022-04-06

I'm trying to solve the following limit:

$\underset{(x,y)\to (2,1)}{lim}\frac{xy{(x-2)}^{\frac{5}{3}}{({y}^{2}-1)}^{\frac{2}{3}}}{\left|y\right|{(x-2)}^{2}+{x}^{2}{(y-1)}^{2}}$

szkliwiakinrd

Beginner2022-04-07Added 10 answers

Try writing

$\underset{(x,y)\to (2,1)}{lim}\frac{{(x-2)}^{\frac{5}{3}}{({y}^{2}-1)}^{\frac{2}{3}}}{{(x-2)}^{2}+{(y-1)}^{2}}$

with the substitution$x\mapsto (u+2)$ and $y\mapsto (v+1)$ as

$\underset{(u,v)\to (0,0)}{lim}\frac{{u}^{\frac{5}{3}}{v}^{\frac{2}{3}}{(v+2)}^{\frac{2}{3}}}{{u}^{2}+{v}^{2}}$

we know that

$u}^{2}\le {u}^{2}+{v}^{2$ (1)

and that

$2uv\le {u}^{2}+{v}^{2}$

Multiply (1) to the 1/2 power times (2) to the 2/3 power to get

$2}^{\frac{2}{3}}{u}^{\frac{5}{3}}{v}^{\frac{2}{3}}\le {({u}^{2}+{v}^{2})}^{\frac{7}{6}$

So the whole fraction is$\le {({u}^{2}+{v}^{2})}^{\frac{1}{6}}$ and that $\to 0$

with the substitution

we know that

and that

Multiply (1) to the 1/2 power times (2) to the 2/3 power to get

So the whole fraction is

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