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Q. 56

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Found in: Page 108

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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# Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\mathrm{lim}}f\left(x\right)=L$ in Exercises $47-60$.$\underset{x\to 2}{\mathrm{lim}}\left(3{x}^{2}-12x+15\right)=3$.

Delta-epsilon proof for $\underset{x\to 2}{\mathrm{lim}}\left(3{x}^{2}-12x+15\right)=3$ is, whenever $0<\left|x-2\right|<\delta$, we also have localid="1648022427369" $\left|\left(3{x}^{2}-12x+15\right)-3\right|<\epsilon$.

See the step by step solution

## Step 1. Given information

$\underset{x\to 2}{\mathrm{lim}}\left(3{x}^{2}-12x+15\right)=3$.

## Step 2. Consider src="data:image/svg+xml;base64,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" localid="1648022623763" ε>0,choose src="data:image/svg+xml;base64,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" localid="1648022634213" δ=ε3.

For all $x$ with $0<\left|x-2\right|<\delta$, we also have localid="1648022643922" $\left|\left(3{x}^{2}-12x+15\right)-3\right|<\epsilon$.localid="1648022660855" $\left|\left(3{x}^{2}-12x+15\right)-3\right|=\left|3{x}^{2}-12x+15-3\right|\phantom{\rule{0ex}{0ex}}=\left|3{x}^{2}-12x+12\right|\phantom{\rule{0ex}{0ex}}=\left|3\left({x}^{2}-4x+4\right)\right|\phantom{\rule{0ex}{0ex}}=3\left|{\left(x-2\right)}^{2}\right|\phantom{\rule{0ex}{0ex}}=3{\left|x-2\right|}^{2}\phantom{\rule{0ex}{0ex}}<3{\left(\delta \right)}^{2}\phantom{\rule{0ex}{0ex}}=3{\left(\sqrt{\frac{\epsilon }{3}}\right)}^{2}\phantom{\rule{0ex}{0ex}}=\epsilon$

Therefore, whenever $0<\left|x-2\right|<\delta$, we also have localid="1648022671089" $\left|\left(3{x}^{2}-12x+15\right)-3\right|<\epsilon$.

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