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Math Help - continuity and limits

  1. #1
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    continuity and limits

    consdier the function:

    f(x) = (sin3x)/x , x<0
    a + 1 , x=0
    be^(-2x), x>0

    a.)explain why f(x) is continuous for all real x =/ 0 (=/ mean does not equal..sry dunno how to type the sign)

    b.)using limits, determine the values of a and b that make f(x) continuous at x = 0

    plz help!
    thx
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  2. #2
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    Quote Originally Posted by ssdimensionss View Post
    consdier the function:

    f(x) = (sin3x)/x , x<0
    a + 1 , x=0
    be^(-2x), x>0

    a.)explain why f(x) is continuous for all real x =/ 0 (=/ mean does not equal..sry dunno how to type the sign)

    b.)using limits, determine the values of a and b that make f(x) continuous at x = 0

    plz help!
    thx
    a.) Use the known properties of each function.

    b.) You need to solve:

    1. \lim_{x \rightarrow 0^-} \frac{\sin(3x)}{x} = a + 1 \Rightarrow 3 = a + 1.

    2. \lim_{x \rightarrow 0^+} b e^{-2x} = a + 1 \Rightarrow b = a + 1.

    I hope calculating the two limits is routine for you.
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  3. #3
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    how do u solve the limit x app 0 sin(3x)/x ? coz if i use sandwich rule the limit does not exist?
    thx
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  4. #4
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    Quote Originally Posted by ssdimensionss View Post
    how do u solve the limit x app 0 sin(3x)/x ? coz if i use sandwich rule the limit does not exist?
    thx
    \lim_{x \rightarrow 0} \frac{\sin (3x)}{x} = {\color{red}3} \lim_{x \rightarrow 0} \frac{\sin (3x)}{({\color{red}3}x)}

    Substitute t = 3x:

    = 3 \lim_{t \rightarrow 0} \frac{\sin t}{t} = 3 (1) = 3.

    It would be expected that you know \lim_{t \rightarrow 0} \frac{\sin t}{t} = 1 and that you can do basic algebraic manipulations of this limit.
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