"I'm trying to solve the following limit:lim(x,y)→(2,1)xy(x−2)53(y2−1)23|y|(x−2)2+x2(y−1)2" was answered by our experts

zdebe5l8

Answered question

2022-04-06

I'm trying to solve the following limit:

lim_{(x, y) \to (2, 1)} \frac{x y {(x - 2)}^{\frac{5}{3}} {(y^{2} - 1)}^{\frac{2}{3}}}{| y | {(x - 2)}^{2} + x^{2} {(y - 1)}^{2}}

szkliwiakinrd

Beginner2022-04-07Added 10 answers

Try writing

lim_{(x, y) \to (2, 1)} \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)

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

we know that

u^{2} \leq u^{2} + v^{2}

(1)
and that

2 u v \leq 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}} \leq {(u^{2} + v^{2})}^{\frac{7}{6}}

So the whole fraction is

\leq {(u^{2} + v^{2})}^{\frac{1}{6}}

and that

\to 0

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