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Q. 48

Expert-verifiedFound in: Page 108

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\mathrm{lim}}f\left(x\right)=L$ in Exercises $47-60$.

$\underset{x\to 2}{\mathrm{lim}}\left(3-4x\right)=-5$.

Delta-epsilon proof for $\underset{x\to 2}{\mathrm{lim}}\left(3-4x\right)=-5$ is, whenever localid="1648006806212" $0<\left|x-2\right|<\delta $, we also have localid="1648021669064" $\left|\left(3-4x\right)+5\right|<\epsilon $.

$\underset{x\to 2}{\mathrm{lim}}\left(3-4x\right)=-5$.

For all $x$ with localid="1648006934128" $0<\left|x-2\right|<\delta $,we also have localid="1648021795730" $\left|\left(3-4x\right)+5\right|<\epsilon $.

localid="1648021807616" $\left|\left(3-4x\right)+5\right|=\left|3-4x+5\right|\phantom{\rule{0ex}{0ex}}=\left|8-4x\right|\phantom{\rule{0ex}{0ex}}=4\left|2-x\right|\phantom{\rule{0ex}{0ex}}=4\left|x-2\right|\phantom{\rule{0ex}{0ex}}<4\delta \phantom{\rule{0ex}{0ex}}=4\left(\frac{\epsilon}{4}\right)\phantom{\rule{0ex}{0ex}}=\epsilon $

Therefore, whenever $0<\left|x-2\right|<\delta $, we also have localid="1648021825608" $\left|\left(3-4x\right)+5\right|<\epsilon $.

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