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Thread: Relationship between left sided and right sided limithow do i prove this ?

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    Relationship between left sided and right sided limithow do i prove this ?

    Assume f is defined for all x near a except possibly at a. Then the lim(x->a+)f(x)=L if snd only if lim(x->a)f(x)=L and lim(x->a-)=L
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    Re: Relationship between left sided and right sided limithow do i prove this ?

    Quote Originally Posted by DiscreteMathHelp View Post
    Assume f is defined for all x near a except possibly at a. a Then the lim(x->a+)f(x)=L if and only if lim(x->a)f(x)=L and lim(x->a-)=L
    First, a bit of notation. $f(a+)$ stands for the right-hand limit of $f$ at $a$, likewise $f(a-)$ is the left-hand limit of $f$ at $a$.
    In this problem it is given that $f(a-)=f(a+)=L$

    Next, if $c>0$ then if $|y-x|<c$ then $x-c<y<x+c$ OR $y\in (x-c,x+c)$. That is $y<x\text{ or }y=x,\text{ or }y<x$.
    So if $x\ne y$ then $x$ is to the right of $y$ OR $x$ is to the left of $y$

    By definition, if $c>0$ and $f(a+)=L$ means then $\exists d>0$ so that if $a<x<a+d$ then $|f(x)-L|<c$.
    Likewise, if $c>0$ and $f(a-)=L$ means that then $\exists d>0$ so that if $a-d<x<a$ then $|f(x)-L|<c$.

    What is the definition of $L$ is the $\displaystyle\lim_{x\to a}f(x)=L$ is
    1. $x\ne a$
    2. if $c>0$ then $\exists d>0$ so that
    3. whenever $0<|x-a|<d$ implies $|f(x)-L|<c$


    Can you see the each definition implies the other?
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    Re: Relationship between left sided and right sided limithow do i prove this ?

    no i need to prove in 2 steps so;
    lim(x->a+)f(x)=L if lim(x->a)f(x)=L and lim(x->a-)=L
    lim(x->a)f(x)=L and lim(x->a-)=L if lim(x->a+)f(x)=L

    how would i do this ? sorry not quite understanding
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    Re: Relationship between left sided and right sided limithow do i prove this ?

    Quote Originally Posted by DiscreteMathHelp;922358[COLOR="#FF0000"
    ]no i need to prove in 2 steps so;[/COLOR]
    lim(x->a+)f(x)=L if lim(x->a)f(x)=L and lim(x->a-)=L
    lim(x->a)f(x)=L and lim(x->a-)=L if lim(x->a+)f(x)=L

    how would i do this ? sorry not quite understanding
    Well you are getting a complete solution from me.
    I will tell you that what needs doing. You are responsible for doing something.
    1) assume that $f(a+)~\&~f(a-)$ both exist. Use those facts and show that the definition of limit at $a$ is satisfied.

    2) Assume that the limit at $a$ exists then argue that implies that both $f(a-)~\&~f(a+) must exist.

    I gladly read your work and offer help then and only then.
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