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Answers without the blur. Sign up and see all textbooks for free! Q. 56

Expert-verified 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 $47-60$.$\underset{x\to 2}{\mathrm{lim}}\left(3{x}^{2}-12x+15\right)=3$.

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

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

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" $\left|\left(3{x}^{2}-12x+15\right)-3\right|<\epsilon$. ### Want to see more solutions like these? 