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Math Help - What is a Limit

  1. #1
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    Exclamation What is a Limit

    I have read the section of my text a couple times and still don't understand, how can I tell whether or not a function has a limit just by looking at it's associated graph
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  2. #2
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    Basically, a limit is what it sounds like. A function f(x) has a limit, say b, at x = a if f(x) become closer and closer to b as x becomes closer to a.

    The value of f(x) at x = a is irrelevant to whether f has a limit at a. However, if f does have a limit b, then putting f(a) = b makes f continuous at a. So, if f cannot be made continuous at a, it does not have a limit.

    Another way to visualize this is to say that you can zoom indefinitely at x = a and still fit the graph of f in some neighborhood of x = a inside your screen. If, at some point, the graph goes through the top or bottom of the screen regardless of how small you make the segment around x = a, the function does not have a limit at a.
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  3. #3
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    One important addition to the above reply is:
    when we say that x is close to a we understand that x\ne a.

    So if \displaystyle\lim _{x \to a} f(x) = b then we know for any x \approx a\;\& \,x \ne a we must have f(x)\approx b.
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  4. #4
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    L is limit of function f when x approach a iff \forall \epsilon >0, \exists \delta>0 \, such \,that\, |x-a|<\delta\Rightarrow |f(x)-L|<\epsilon.

    if x in delta neighbourhood of a then f(x) in epsilon neighbourhood of L
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  5. #5
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    Quote Originally Posted by MathoMan View Post
    L is limit of function f when x approach a iff \forall \epsilon >0, \exists \delta>0 \, such \,that\, |x-a|<\delta\Rightarrow |f(x)-L|<\epsilon.
    if x in delta neighbourhood of a then f(x) in epsilon neighbourhood of L
    @MathoMan
    You have miss a crucial bit in the definition.

    \forall \epsilon >0, \exists \delta>0 \text{ such that }0< |x- a|<\delta\Rightarrow |f(x)-L|<\epsilon.
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