Find the following limit without using L’Hopital’s rule: lim x → 1 x m − 1 x n − 1

Question

Find the following limit without using L’Hopital’s rule:      lim          x      →      1                          x        m            −      1                      x        n            −      1

zwichtsu · Accepted Answer

We note that:      ∑          k      =      0        n        x    k    =                    x                  n          +          1                    −      1              x      −      1      As such, we have:                    ∑                  k          =          0                          m          −          1                            x        k                            ∑                  k          =          0                          n          −          1                            x        k              =            (                                    x                          m                                −          1                          x          −          1                    )              (                                    x                          n                                −          1                          x          −          1                    )        =                    x        m            −      1                      x        n            −      1      As such,       lim          x      →      1                          x        m            −      1                      x        n            −      1        =      lim          x      →      1                          ∑                  k          =          0                          m          −          1                            x        k                            ∑                  k          =          0                          n          −          1                            x        k              =                    ∑                  k          =          0                          m          −          1                            1        k                            ∑                  k          =          0                          n          −          1                            1        k              =                                          m            n

Monfredo0n · Accepted Answer

You can use that       ∑          k      =      0              m      −      1            x    k    =                    x        m            −      1              x      −      1      . Then you have      lim          x      →      1                          x        m            −      1                      x        n            −      1        =      lim          x      →      1                  x      −      1              x      −      1        ⋅                    ∑                  k          =          0                          m          −          1                            x        k                            ∑                  k          =          0                          n          −          1                            x        k

Find the following limit without using L’Hopital’s rule: <munder> <mo movablelimits="true" f

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