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Math Help - Function with no limit at point c

  1. #1
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    Function with no limit at point c

    Find a case of the function f and g where f and g do not have limits at a point c, but f+g and fg have limits at c.


    Would

    f(x)=(-1)^x
    -and-
    g(x)=(-1)^{x+1}

    work? if x>1
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  2. #2
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    What point c were you thinking of for your choice of f(x) and g(x)?

    How about f(x) = 0 if x is rational; 1 if x is irrational. g(x) = 1 if x is rational; 0 if x is irrational. Then f(x) + g(x) = 1, f(x)g(x) = 0, and the limit does not exist for f(x) or for g(x) at any point c.
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  3. #3
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    Just a general point C.
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  4. #4
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    Don't complicate things.
    f(x) = \left\{ {\begin{array}{rl}<br />
   1 & {x > 0}  \\<br />
   { - 1} & {x < 0}  \\<br /> <br />
 \end{array} } \right.\quad \& \quad g(x) = \left\{ {\begin{array}{rl}<br />
   { - 1} & {x > 0}  \\<br />
   1 & {x < 0}  \\<br /> <br />
 \end{array} } \right.
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  5. #5
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    ok so there exists a limit at 3, but not at 0. I guess where I was getting confused is that I though c must be arbitrary so it had to go for all points.
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  6. #6
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    Well O.K. Make it:
    f(x) = \left\{ {\begin{array}{rl}<br />
   1 & {x > c}  \\<br />
   { - 1} & {x < c}  \\<br /> <br />
 \end{array} } \right.\quad \& \quad g(x) = \left\{ {\begin{array}{rl}<br />
   { - 1} & {x > c}  \\<br />
   1 & {x < c}  \\<br /> <br />
 \end{array} } \right.

    Does that work? WHY or WHY NOT?
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