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Found in: Page 108

### Calculus

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 $47-60$.$\underset{x\to -3}{\mathrm{lim}}\left(1-x\right)=4$.

Delta-epsilon proofs for $\underset{x\to -3}{\mathrm{lim}}\left(1-x\right)=4$ is, whenever $0<\left|x+3\right|<\delta$, we also have localid="1648022140552" $\left|\left(1-x\right)-4\right|<\epsilon$.

See the step by step solution

## Step 1. Given information

$\underset{x\to -3}{\mathrm{lim}}\left(1-x\right)=4$.

For all $x$ with $0<\left|x+3\right|<\delta$, we also have localid="1648022193077" $\left|\left(1-x\right)-4\right|<\epsilon$.

localid="1648022202619" $\left|\left(1-x\right)-4\right|=\left|1-x-4\right|\phantom{\rule{0ex}{0ex}}=\left|-x-3\right|\phantom{\rule{0ex}{0ex}}=\left|-\left(x+3\right)\right|\phantom{\rule{0ex}{0ex}}=\left|x+3\right|\phantom{\rule{0ex}{0ex}}<\delta \phantom{\rule{0ex}{0ex}}=\epsilon$

Therefore, whenever $0<\left|x+3\right|<\delta$, we also have localid="1648022212066" $\left|\left(1-x\right)-4\right|<\epsilon$.