Question

Compute the limit $\lim_{(x,y)\to(0,0)} \frac{x^3y^2 + 2y^3}{x^3 + y^3}$ or prove that it does not exist.

Name: compute the limit limxyto00 fracx3y2 2y3x3 y3 or prove that it does not exist 06899
Uploaded: 2018-01-20T01:29:46-08:00
Duration: 33 s
Channel: Alan Ghazarians
Description: compute the limit limxyto00 fracx3y2 2y3x3 y3 or prove that it does not exist 06899

          Compute the limit $\lim_{(x,y)\to(0,0)} \frac{x^3y^2 + 2y^3}{x^3 + y^3}$ or prove that it does not exist.

$compute the limit limxyto00 fracx3y2 2y3x3 y3 or prove that it does not exist 06899$

Added by Diana J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Alan Ghazarians

01/20/2018

Step 1

We have $$ L = \lim_{(x,y)\to(0,0)} \frac{x^3y^2 + 2y^3}{x^3 + y^3} $$ Let us approach $(0,0)$ along the path $y = mx$. Then \begin{align*} \label{eq:1} L &= \lim_{x\to 0} \frac{x^3(mx)^2 + 2(mx)^3}{x^3 + (mx)^3} \\ &= \lim_{x\to 0} \frac{m^2x^5 + 2m^3x^3}{x^3(1 + Show more…

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