How can I find the constants A and B if lim x → ∞ ( x 2 + x − A x − B ) = 0 ?

Question

How can I find the constants A and B if      lim          x      →      ∞            (                  x        2            +      x        −    A    x    −    B    )    =  0          ?

Judovh0 · Accepted Answer

You&#039;re trying to find the asymptote of             x      2        +    x   at infinity. The standard method is to say that if      lim          x      →      ∞                  x      2        +    x    −  A  x  −  B  =  0then surely also      lim          x      →      ∞                                    x          2                +        x            −      A      x      −      B        x    =  0which implies  A  =      lim          x      →      ∞                          x        2            +      x        x    =      lim          x      →      ∞            1    +          1      x        =  1So   A  =  1 and now you can substitute this back into the original equation to find B.

Summer Bradford · Accepted Answer

(  (  x  +  1      /    2      )    2    −  1      /    4      )          1              /            2        −  A  x  −  B  ;Set   y  :=  x  +  1      /    2  , and consider   y  →  ∞  (      y    2    −  1      /    4      )          1              /            2        −  A  (  y  −  1      /    2  )  −  B  ;The asymptote of   (      y    2    −  1      /    4      )          1              /            2       is y.Collecting corresponding terms in the asymptotic expression:  y  (  1  −  A  )  +  (  (  1      /    2  )  A  −  B  )  =  0 we get  A  =  1 and   B  =  1      /    2.

How can I find the constants A and B if <munder> <mo movablelimits="true" form="prefix">lim

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