how am i find the values of?

Discussion in 'Differentiation and Integration' started by remy, Nov 12, 2021.

  1. remy

    remy

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    how to i find what a,b or c by myself?

    the question is what their values if the function is
    Continuous

    thanks :)
     

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    remy, Nov 12, 2021
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  2. remy

    HallsofIvy

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    You start by knowing what "continuous" means! If f is "continuous" then it is continuous at all x and, in particular at x= -1. In order that f be continuous at x= -1, the limit, as x goes to -1 must be f(-1)= -3.

    But for x anything other than -1, f(x) is a fraction with (x+ 1)^2 in the denominator. In order that the limit exist, the numerator must also contain a factor of (x+ 1)^2 to cancel that. The numerator is cubic so it must be of the form (x+ 1)^2(px+ q)= (x^2+ 2x+ 1)(px+ q)= px^3+ qx^2+ 2px^2+ 2qx+ px+ q= px^3+ (q+ 2p)x^2+ (2q+ p)x+ q

    That must be equal to x^3+ ax^2+ bx+ c so p= 1 and we then have a=q+ 2, b= 2q+ 1, and c= q.

    Of course, canceling the (x+ 1)^2 leaves px+ q= x+ q and the limit of that, as x goes to -1 is -1+ q= 3 so q= 4.
     
    HallsofIvy, Nov 12, 2021
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