1-7. The Precise Definition of a limit

2. Definition Let $f$ be a function on some open interval that contains the number $a$ except possibly at $a$ itself. Then we say that the limit of $f(x)$ as $x$ approaches $a$ is $L$, and we write

$\displaystyle \lim_{x\rightarrow a}f(x)=L$

if for every number $\epsilon > 0$ there is a number $\delta > 0$ such that
if $\displaystyle 0< \left |x-a \right |<\delta$ then $\displaystyle \left |f(x)-L \right |<\epsilon$

3. Definition of Left-Hand Limit

$\displaystyle \lim_{x\rightarrow a^{-}}f(x)=L$

if for every number $\displaystyle \epsilon > 0$ there is a number $\displaystyle \delta > 0$ such that

if $\displaystyle a-\delta then $\displaystyle \left | f(x)-L \right |< \epsilon$

4. Definition of Right-Hand Limit

$\displaystyle \lim_{x\rightarrow a^{+}}f(x)=L$

if for every number $\displaystyle \epsilon > 0$ there is a number $\displaystyle \delta> 0$ such that
if $\displaystyle a then $\displaystyle \left | f(x)-L \right |< \epsilon$

6- Definition Let $\displaystyle f$ be a function defined on some open interval that contains the number $\displaystyle a$ except possibly at $\displaystyle a$ itself. Then
$\displaystyle \lim_{x\rightarrow a}f(x)=\infty$
means that for every positive number $\displaystyle M$ there is a positive number $\displaystyle \delta$ such that
if $\displaystyle 0< \left | x-a \right |< \delta$ then $\displaystyle f(x)> M$

7. Definition let $\displaystyle f$ be a function defined on some open interval that contains the number $\displaystyle a$ except possibly at $\displaystyle a$ itself. then
$\displaystyle \lim_{x\rightarrow a}f(x)=-\infty$
means that for every negative number $\displaystyle N$ there is a positive number $\displaystyle \delta$ such that
if $\displaystyle 0< \left | x-a \right |< \delta$ then $\displaystyle f(x)< N$
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