I think it's obvious but I cannot show it explicitly.If a function f is continuous and positive definite, that is, f ( 0 ) = 0 and f ( x ) &gt; 0 for all nonzero x.Then, how to show that lim t → ∞ f ( x t ) = 0 ⟹ lim t → ∞ x t = 0?

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I think it&#039;s obvious but I cannot show it explicitly.If a function   f is continuous and positive definite, that is,   f  (  0  )  =  0 and   f  (  x  )  &amp;gt;  0 for all nonzero   x.Then, how to show that       lim          t      →      ∞        f  (      x    t    )  =  0    ⟹        lim          t      →      ∞            x    t    =  0?

Brooks Butler · Accepted Answer

The result is false. Think of   f  (  x  )  =                    x        2                    1        +                  x          4                    . You have that      lim          n      →      ∞        f  (  n  )  =  0  ,but       lim          n      →      ∞        n  ≠  0.

I think it's obvious but I cannot show it explicitly. If a function f is continuous and posi

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