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Q. 9

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.$\underset{\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\to \left(1,0,-1\right)}{\mathrm{lim}}\frac{\mathrm{sin}\left(xy\right)}{{\mathrm{x}}^{2}-{\mathrm{y}}^{2}+{\mathrm{z}}^{2}}$

$\underset{\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\to \left(1,0,-1\right)}{\mathrm{lim}}\frac{\mathrm{sin}\left(xy\right)}{{\mathrm{x}}^{2}-{\mathrm{y}}^{2}+{\mathrm{z}}^{2}}=0$

See the step by step solution

## Step 1. Given information is:

$\underset{\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\to \left(1,0,-1\right)}{\mathrm{lim}}\frac{\mathrm{sin}\left(\mathrm{xy}\right)}{{\mathrm{x}}^{2}-{\mathrm{y}}^{2}+{\mathrm{z}}^{2}}$

## Step 2. Evaluating Limits

$\mathrm{Since},\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\to \left(1,0,-1\right)\phantom{\rule{0ex}{0ex}}\mathrm{sin}\left(\mathrm{xy}\right)\to 0\mathrm{and}{\mathrm{x}}^{2}-{\mathrm{y}}^{2}+{\mathrm{z}}^{2}\to 2\mathrm{which}\mathrm{is}\mathrm{finite}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{So}\mathrm{the}\mathrm{function}\frac{\mathrm{sin}\left(\mathrm{xy}\right)}{{\mathrm{x}}^{2}-{\mathrm{y}}^{2}+{\mathrm{z}}^{2}}\mathrm{is}\mathrm{continuous}\mathrm{at}\left(1,0,-1\right).\phantom{\rule{0ex}{0ex}}\mathrm{So},\phantom{\rule{0ex}{0ex}}\underset{\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\to \left(1,0,-1\right)}{\mathrm{lim}}\frac{\mathrm{sin}\left(\mathrm{xy}\right)}{{\mathrm{x}}^{2}-{\mathrm{y}}^{2}+{\mathrm{z}}^{2}}=\frac{\mathrm{sin}\left(0\right)}{{1}^{2}-{0}^{2}+{\left(-1\right)}^{2}}=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\underset{\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\to \left(1,0,-1\right)}{\mathrm{lim}}\frac{\mathrm{sin}\left(\mathrm{xy}\right)}{{\mathrm{x}}^{2}-{\mathrm{y}}^{2}+{\mathrm{z}}^{2}}=0$