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Q. 44

Expert-verifiedFound in: Page 107

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

For each limit statement in Exercises $41-44$, use algebra to find $\delta $ or $N$ in terms of $\epsilon $ or $M$, according to the appropriate formal limit definition.

$\underset{x\to \infty}{\mathrm{lim}}\left({x}^{2}+2\right)=\infty $,find $N$ in terms of $M$.

For $\underset{x\to \infty}{\mathrm{lim}}\left({x}^{2}+2\right)=\infty $, $N=\sqrt{M-2}$.

$\underset{x\to \infty}{\mathrm{lim}}\left({x}^{2}+2\right)=\infty $.

$f\left(x\right)={x}^{2}+2\phantom{\rule{0ex}{0ex}}c=\infty \phantom{\rule{0ex}{0ex}}L=\infty $

Now for $\delta >0$.

$\left|x\right|<N\phantom{\rule{0ex}{0ex}}{\left|x\right|}^{2}<{N}^{2}\phantom{\rule{0ex}{0ex}}\left|{x}^{2}+2\right|<{N}^{2}+2\phantom{\rule{0ex}{0ex}}\left|{x}^{2}+2\right|<M$

Therefore,

${N}^{2}+2=M\phantom{\rule{0ex}{0ex}}{N}^{2}=M-2\phantom{\rule{0ex}{0ex}}N=\sqrt{M-2}$

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