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Q. 33

Expert-verifiedFound in: Page 107

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

For each limit statement , use algebra to find δ > 0 in terms of $\epsilon $ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $\epsilon $.

$\underset{x\to 0}{\mathrm{lim}}(5{x}^{2}-1)=-1$

$\delta =\sqrt{\frac{\epsilon}{5}}$

We have been given a limit statement as $\underset{x\to 0}{\mathrm{lim}}(5{x}^{2}-1)=-1$.

We have to find $\delta \mathrm{in}\mathrm{terms}\mathrm{of}\epsilon $.

From the given limit statement, we can identify

$f\left(x\right)=5{x}^{2}-1\phantom{\rule{0ex}{0ex}}c=0\phantom{\rule{0ex}{0ex}}L=-1\phantom{\rule{0ex}{0ex}}\mathrm{For}\epsilon >0\phantom{\rule{0ex}{0ex}}\left|\left(5{x}^{2}-1\right)-(-1)\right|<\epsilon \phantom{\rule{0ex}{0ex}}\left|5{x}^{2}-1+1\right|<\epsilon \phantom{\rule{0ex}{0ex}}\left|5{x}^{2}\right|<\epsilon \phantom{\rule{0ex}{0ex}}5\left|{x}^{2}\right|<\epsilon \phantom{\rule{0ex}{0ex}}\left|{x}^{2}\right|<\frac{\epsilon}{5}\phantom{\rule{0ex}{0ex}}\left|x\right|<\sqrt{\frac{\epsilon}{5}}\phantom{\rule{0ex}{0ex}}For0<|x-0|<\delta ,weget\left|x\right|<\sqrt{\frac{\epsilon}{5}}\phantom{\rule{0ex}{0ex}}\mathrm{Therefore},\delta =\sqrt{\frac{\epsilon}{5}}$

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