How to find the limit of x 7 x as x approaches infinity?

Question

How to find the limit of             x                        7          x                     as x approaches infinity?

chaghirun · Accepted Answer

lim              x        →        ∞                    x                        7          x                      =    1   Solution:First, we will use the following:            e              ln                  (          x          )                      =    x  Because             e      x       is continuous on             (      -      ∞      ,      ∞      )      , we have             lim              x        →        ∞                    e              f                  (          x          )                      =          e                        lim                      x            →            ∞                                    f                      (            x            )                              With these:            lim              x        →        ∞                    x                        7          x                      =          lim              x        →        ∞                    e                        ln                      (                          x                                                7                  x                                                      )                                    =          lim              x        →        ∞                    e                        7          x                          ln                      (            x            )                                    =          e                        lim                      x            →            ∞                                    7          x                          ln                      (            x            )                              Therefore, we will use L&#039;Hopital&#039;s rule:            lim              x        →        ∞                    7      x              ln              (        x        )              =          lim              x        →        ∞                            7                  ln                      (            x            )                              x            =          lim              x        →        ∞                                      d                      d            x                          7                  ln                      (            x            )                                                d                      d            x                          x                  =          lim              x        →        ∞                                      7          x                    1            =          lim              x        →        ∞                    7      x            =    0  When we combine these, we get the following final result:            lim              x        →        ∞                    x                        7          x                      =          e                        lim                      x            →            ∞                                    7          x                          ln                      (            x            )                                =          e      0        =    1

How do you find the limit of x^(7/x) as x approaches infinity?

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