How to evaluate the limit lim e t t as t → ∞ ?

Question

How to evaluate the limit       lim                  e        t            t       as       t    →    ∞  ?

pomorijujfo · Accepted Answer

Method 1: L&#039;Hôpital&#039;s RuleThe limit:            lim              t        →        ∞                            e        t            t      is of an indeterminate form             ∞      ∞      , and so we can apply L&#039;Hôpital&#039;s rule which states that for an indeterminate limit then, providing the limits exits then:            lim              x        →        a                            f                  (          x          )                            g                  (          x          )                      =          lim              x        →        a                            f        ′                  (          x          )                            g        ′                  (          x          )                    Applying L&#039;Hôpital&#039;s rule, we thus arrive at:            lim              t        →        ∞                            e        t            t        =          lim              t        →        ∞                                      d                      d            t                                    e          t                                      d                      d            t                          t                                                        =          lim              t        →        ∞                                      e          t                            1             = oo  as             e      t       is unbounded.Method 2: Graphicallygraph{(e^x)/x [-26.97, 23.66, -2.97, 22.35]} Nothing more to say; clearly       y    =                  e        x            x       is unboundedMethod 3: Taylor Series                    e        t            t        =          1      t                   e      t                                =          1      t              {      1      +      t      +                        t          2                          2          !                    +                        t          3                          3          !                    +                        t          4                          4          !                    +      ...      }                                =          1      t        +    1    +          t              2        !              +                  t        2                    3        !              +                  t        3                    4        !              +    ...  So then:            lim              t        →        ∞                            e        t            t        =          lim              t        →        ∞                    {              1        t            +      1      +              t                  2          !                    +                        t          2                          3          !                    +                        t          3                          4          !                    +      ...      }                                                 =          lim              t        →        ∞                    {              1        t            +      1      +      O              (        t        )            }                                                 =          lim              t        →        ∞                    {              1        t            +      1      }        +          lim              t        →        ∞                    {      O              (        t        )            }

How to evaluate the limit lim e^t/t as t->oo?

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