I'm wondering about the motivation for proving the limit. lim x → 1 x n = 1

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I&#039;m wondering about the motivation for proving the limit.            lim              x        →        1                    x      n        =    1

xxsailojaixxv5 · Accepted Answer

Lets go to the definition of the limit.  ∀  ϵ  &amp;gt;  0  ,  ∃  δ  &amp;gt;  0  :      |    x  −  1      |    &amp;lt;  δ    ⟹        |        x    n    −  1      |    &amp;lt;  ϵThe &quot;trick&quot; here will be very similar to the one above, but it might look more natural here.Multiplying by &quot;1.&quot;      |                            x          n                −        1            1                      1        +                  x          n                +                              x            n                    2                +        ⋯        +                              x            n                                n            −            1                                      1        +                  x          n                +                              x            n                    2                +        ⋯        +                              x            n                                n            −            1                                |    =      |                  x        −        1                    1        +                  x          n                +                              x            n                    2                +        ⋯        +                              x            n                                n            −            1                                |  let   δ  ≤  1 then   x  &amp;gt;  0 and   1  +      x    n    +            x      n        2    +  ⋯  +            x      n              n      −      1        &amp;gt;  1      |        x    n    −  1      |    &amp;lt;      |    x  −  1      |    &amp;lt;  δ

I'm wondering about the motivation for proving the limit. <mstyle displaystyle="true" scriptleve

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