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Q. 43

Expert-verified
Found in: Page 107

### Calculus

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.$\underset{x\to {1}^{-}}{\mathrm{lim}}\frac{1}{1-x}=\infty$, find $\delta$ in terms of $M$.

For $\underset{x\to {1}^{-}}{\mathrm{lim}}\frac{1}{1-x}=\infty$, $\delta =\frac{1}{M}$.

See the step by step solution

## Step 1. Given information

$\underset{x\to {1}^{-}}{\mathrm{lim}}\frac{1}{1-x}=\infty$.

## Step 2. From the limit expression,

$f\left(x\right)=\frac{1}{1-x}\phantom{\rule{0ex}{0ex}}c=1\phantom{\rule{0ex}{0ex}}L=\infty$

Now for $\delta >0$.

$\left|x-1\right|<\delta \phantom{\rule{0ex}{0ex}}\left|1-x\right|<\delta \phantom{\rule{0ex}{0ex}}\frac{1}{\left|1-x\right|}<\delta \phantom{\rule{0ex}{0ex}}\left|\frac{1}{1-x}\right|<\delta \phantom{\rule{0ex}{0ex}}

Hence, $\delta =\frac{1}{M}$.