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Q. 7

Expert-verified
Found in: Page 989

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# Evaluate the following limits, or explain why the limit does not exist.$\underset{\left(x,y\right)\to \left(0,0\right)}{\mathrm{lim}}\frac{{x}^{3}+{y}^{3}}{{x}^{2}+{y}^{2}}$

$\underset{\left(\mathrm{x},\mathrm{y}\right)\to \left(0,0\right)}{\mathrm{lim}}\frac{{\mathrm{x}}^{3}+{\mathrm{y}}^{3}}{{\mathrm{x}}^{2}+{\mathrm{y}}^{2}}=0$

See the step by step solution

## Step 1. Given information is:

$\underset{\left(\mathrm{x},\mathrm{y}\right)\to \left(0,0\right)}{\mathrm{lim}}\frac{{\mathrm{x}}^{3}+{\mathrm{y}}^{3}}{{\mathrm{x}}^{2}+{\mathrm{y}}^{2}}$

## Step 2. Evaluating Limits

$\mathrm{Since},\left(\mathrm{x},\mathrm{y}\right)\to \left(0,0\right)\phantom{\rule{0ex}{0ex}}{\mathrm{x}}^{3}+{\mathrm{y}}^{3}=0\mathrm{and}{\mathrm{x}}^{2}+{\mathrm{y}}^{2}=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{So}\mathrm{to}\mathrm{find}\underset{\left(\mathrm{x},\mathrm{y}\right)\to \left(0,0\right)}{\mathrm{lim}}\frac{{\mathrm{x}}^{3}+{\mathrm{y}}^{3}}{{\mathrm{x}}^{2}+{\mathrm{y}}^{2}},\mathrm{take}\mathrm{x}=\mathrm{r}\mathrm{cos\theta }\mathrm{and}\mathrm{y}=\mathrm{r}\mathrm{sin\theta }\phantom{\rule{0ex}{0ex}}\mathrm{then},\left(\mathrm{x},\mathrm{y}\right)\to \left(0,0\right)\mathrm{when}\mathrm{r}\to 0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Now},\frac{{\mathrm{x}}^{3}+{\mathrm{y}}^{3}}{{\mathrm{x}}^{2}+{\mathrm{y}}^{2}}=\frac{{\mathrm{r}}^{3}{\mathrm{cos}}^{3}\mathrm{\theta }+{\mathrm{r}}^{3}{\mathrm{sin}}^{3}\mathrm{\theta }}{{\mathrm{r}}^{2}{\mathrm{cos}}^{2}\mathrm{\theta }+{\mathrm{r}}^{2}{\mathrm{sin}}^{2}\mathrm{\theta }}=\mathrm{r}\left({\mathrm{cos}}^{3}\mathrm{\theta }+{\mathrm{sin}}^{3}\mathrm{\theta }\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Therefore},\phantom{\rule{0ex}{0ex}}\underset{\left(\mathrm{x},\mathrm{y}\right)\to \left(0,0\right)}{\mathrm{lim}}\frac{{\mathrm{x}}^{3}+{\mathrm{y}}^{3}}{{\mathrm{x}}^{2}+{\mathrm{y}}^{2}}=\underset{\mathrm{r}\to 0}{\mathrm{lim}}\left[\mathrm{r}\left({\mathrm{cos}}^{3}\mathrm{\theta }+{\mathrm{sin}}^{3}\mathrm{\theta }\right)\right]=0\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\underset{\left(\mathrm{x},\mathrm{y}\right)\to \left(0,0\right)}{\mathrm{lim}}\frac{{\mathrm{x}}^{3}+{\mathrm{y}}^{3}}{{\mathrm{x}}^{2}+{\mathrm{y}}^{2}}=0$