Calculate this limit using Maclaurin series: lim x → ∞ ( ( x 3 − x 2 + 2 x ) e 1 x − x 3 + x 6 )

Question

Calculate this limit using Maclaurin series:       lim          x      →      ∞        (  (      x    3    −      x    2    +      2    x    )      e                  1        x              −            x      3        +          x      6        )

Laura Fletcher · Accepted Answer

Step 1I&#039;m not entirely sure how the hypothesis about the expectation comes into play, but here&#039;s my solution. I&#039;ll simply use the Markov inequality for a non-negative r.v. and   a  &amp;gt;  0 in the form  E  [  X  ]  ≥  a  P  (  X  ≥  a  )with   a  =  2. Remembering that probabilities are non-negative and using Markov we have:  E  [  X  ]  ≥  P  (  X  ≥  2  )  +  P  (  X  ≥  2  )  ≥  P  (  X  ≥  2  )Edit after the comments about non-negativity:I think I saw the light :D I use the extended form of Markov inequality for monotonically increasing function:Step 2TheoremLet   φ be a monotonically increasing function for   x  &amp;gt;  0 and let X be a r.v.,   a  &amp;gt;  0  ,  φ  (  a  )  &amp;gt;  0. Then  P  (  X  ≥  a  )  ≤                    E            [      φ      (      X      )      ]              φ      (      a      )      In our case I&#039;ll consider the (somehwat artificial) function   φ  (  x  )  :=  x  +      e          −      x        −  1, which we can prove satisfy all the hypothesis with   a  =  2. Hence, since   1      /    φ  (  a  )  ≤  1 and using the bound for       e          −      X      , we have                    P        (        X        ≥        a        )                            ≤                                            E                        [            X            +                          e                              −                X                                      −            1            ]                                φ            (            a            )                                                            ≤                  E                [        X        ]        +                  E                [                  e                      −            X                          ]        −        1                                          ≤                  E                [        X        ]

Calculate this limit using Maclaurin series: lim x rarr oo((x^3−x^2+2/x)e^1/x−sqrt(x^3+x^6))

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