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Math Help - Showing that function is discontinuous in a point

  1. #1
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    Showing that function is discontinuous in a point

    f(x) = sin(pi/x), when x != 0, and 0, when x = 0

    Prove that f(x) is discontinuous in x = 0

    Is it necessary to use the delta-epsilon-definition, or do you think i could sketch the graph and point to the jump in value?

    If I need to use the definition, could someone give me a hint?
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  2. #2
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    As x \to 0, \frac{\pi}{x} \to \infty.

    Since the sine function oscillates, that means \sin{\frac{\pi}{x}} will oscillate between -1 and 1 quicker and quicker as x \to 0.

    Since \sin{\frac{\pi}{x}} will not tend to a single value, that means the limit does not exist and is thus discontinuous.
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