Find the asymptotesFind the asymptotes of f : R → R , f ( x ) = e x − e 2 x + e 4 x ln 2 ⁡ ( 1 + e − x ) 3 .I found that y=0 is an asymptote when x → − ∞, but how do I calculate lim x → ∞ f ( x ) ?

Question

Find the asymptotesFind the asymptotes of   f  :      R    →      R    ,  f  (  x  )  =                    e                  x                    −              e                  2          x                    +              e                  4          x                            ln                  2                    ⁡      (      1      +              e                  −          x                    )        3    .I found that y=0 is an asymptote when   x  →  −  ∞, but how do I calculate       lim          x      →      ∞        f  (  x  ) ?

Brendon Bentley · Accepted Answer

Let see what happens when   x  →  ∞. We have   ln  ⁡  (  1  +  y  )  =  y  −            y      2        2    +            y      3        3    +  O  (      y    4    ) as   y  →  0 hence  f  (  x  )  =            (              e        x            −              e                  2          x                    +              e                  4          x                                      (                      e                          −              x                                −                                    e                              −                2                x                                      2                    +                                    e                              −                3                x                                      3                    +          O          (                      e                          −              4              x                                )          )                2            )              1              /            3      that is  f  (  x  )  =            (              e        x            −              e                  2          x                    +              e                  4          x                            (                  e                      −            2            x                          −                  e                      −            3            x                          +                  11          12                          e                      −            4            x                          +        O        (                  e                      −            5            x                          )        )            )              1              /            3      therefore  f  (  x  )  =            (              11        12            +      O      (              e                  −          x                    )      )              1              /            3        .In particular,      lim          x      →      ∞        f  (  x  )  =            (              11        12            )              1              /            3        .

Find the asymptotes of f:R rightarrow R, f(x)= root(3)(ex−e2x+e4xln2(1+e−x)). I found that y=0 is an asymptote when x rightarrow −infinity, but how do I calculate limx rightarrow infinity f(x) ?

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