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Math Help - Limits - true and false help

  1. #1
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    Limits - true and false help

    I can do limits - but not when it comes to true and false:

    1. if lim X->0 f(x) = 0, then there exists c such that f (c) <.001

    2. f is undefined for x=c, then lim x-> c f (x) does not exist

    3. If lim x->c f (x) = L and f(c) = L then f is continuous at c

    4. If f(x) = g(x) for x not equal to c and f(c) not equal to g(c) then f or g must be discontinuous at c

    5. If f is continuous on (a,b], then f must take on both a maximum and minimum on (a,b]

    6. Rational functions have infinitely many discontinuities.

    7. Trigonometric functions can have infinitely many discontinuities.

    If you can help explain any or all of those to me, I would be thankful.
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  2. #2
    MHF Contributor
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    Talking

    Use the definitions and intuitive meanings of the terms and processes:

    Quote Originally Posted by BooGTS View Post
    1. if lim X->0 f(x) = 0, then there exists c such that...
    ...|f(c) - 0| < (some small value).

    Quote Originally Posted by BooGTS View Post
    2. f is undefined for x=c, then lim x-> c f (x) does not exist
    Is "limit exists at" the same thing as "continuous at"?

    Quote Originally Posted by BooGTS View Post
    3. If lim x->c f (x) = L and f(c) = L then f is continuous at c
    What is the definition of "continuous"?

    Quote Originally Posted by BooGTS View Post
    4. If f(x) = g(x) for x not equal to c and f(c) not equal to g(c) then f or g must be discontinuous at c
    Suppose that f is continuous at c, so that lim, x->c, f(x) = f(c). What does this tell you about g(c)?

    Quote Originally Posted by BooGTS View Post
    5. If f is continuous on (a,b], then f must take on both a maximum and minimum on (a,b]
    Take a close look at the definitions and theorems. Does this half-open interval fulfill the requirements for this conclusion?

    Quote Originally Posted by BooGTS View Post
    6. Rational functions have infinitely many discontinuities.
    Think about a rational function you've graphed recently, like in your algebra class. Did you draw infinitely-many vertical asymptotes?

    Quote Originally Posted by BooGTS View Post
    7. Trigonometric functions can have infinitely many discontinuities.
    Think about the trig functions you've graphed. What do a couple of them look like, with respect to vertical asymptotes?

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