# Chapter 1 - Functions and Limits - 1.7 The Precise Definition of a Limit - 1.7 Exercises - Page 82: 23

#### Work Step by Step

$\displaystyle \lim_{x\rightarrow a}f(x)=L$ if for every number $\epsilon > 0$ there is a number $\delta > 0$ such that the following is valid: $($if $0 < |x-a| < \delta$ then $|f(x)-L| < \epsilon)$ ------------- $f(x)=x.$ Given any $\epsilon > 0$, we want to find a $\delta > 0$ such that $0 < |x-a| < \delta\ \ \Rightarrow\ \ |x-a| < \epsilon$. So we take $\delta=\epsilon ,$ and, by the definition, $\displaystyle \lim_{x\rightarrow a}x=a$

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