Best solutions to - "Determine the limits if they exist: lim(x,y)→(2,4)(x−2)2(y−4)2(x−2)3+(y−4)3"

hexacordoK

Answered question

2020-10-19

Determine the limits if they exist:

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

Answer & Explanation

Ezra Herbert

Skilled2020-10-20Added 99 answers

Given that $lim_{(x, y) \to (2, 4)} \frac{(x - 2)^{2} (y - 4)^{2}}{(x - 2)^{3} + (y - 4)^{3}}$
Take
$(x - 2) = X$ and $4 = Y$ then
when $x \to 2$ then $X \to 0$
when $y \to 4$ , then $Y \to 0$
So, the equation (1) becomes
$lim_{(x, y) - > (2, 4)} \frac{(x - 2)^{2} (y - 4)^{2}}{(x - 2)^{3} + (y - 4)^{3}} = lim_{(X, Y) \to (0, 0)} \frac{x^{2} \cdot y^{2}}{X^{3} \cdot Y^{3}}$
To check is limit path dependent or not, so put Y=mX
$\Rightarrow lim_{(X, Y) - > (0, 0)} \frac{X^{2} \cdot Y^{2}}{X^{3} + Y^{3}} = lim_{X \to 0} \frac{X^{2} \cdot m^{2} \cdot X^{2}}{X^{3} + (m X)^{3}} = lim_{X - > 0} \frac{m^{2} \cdot X^{4}}{X^{3} (1 + m^{3})}$
$\Rightarrow lim_{(X, Y) - > (0, 0)} \frac{X^{2} \cdot Y^{2}}{X^{3} + Y^{3}} = lim_{X \to 0} \frac{m^{2} \cdot X^{4}}{X^{3} (1 + m^{3})} = 0$
$lim_{(x, y) - > (2, 4)} \frac{(x - 2)^{2} (y - 4)^{2}}{(x - 2)^{3} + (y - 4)^{3}} = lim_{(X, Y) - > (0, 0)} \frac{x^{2} \cdot y^{2}}{X^{3} \cdot Y^{3}} = 0$
Hence, limit exist and
$lim_{(x, y) \to (2, 4)} \frac{(x - 2)^{2} (y - 4)^{2}}{(x - 2)^{3} + (y - 4)^{3}} = 0$

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