If f ( x ) = 1 3 ( f ( x + 1 ) + 5 f ( x + 2 ) ) and f ( x ) &gt; 0 x ∈ R then lim x → ∞ f ( x ) is?

Question

If   f  (  x  )  =      1    3              (        f  (  x  +  1  )  +      5          f      (      x      +      2      )                  )       and   f  (  x  )  &amp;gt;  0  x  ∈      R   then       lim          x      →      ∞        f  (  x  ) is?

Kaylyn Ewing · Accepted Answer

To say that       lim          x      →      ∞        f  (  x  )  =  L means that for all   ϵ  &amp;gt;  0 there exists   M  &amp;gt;  0 such that      |    f  (  x  )  −  L      |    &amp;lt;  ϵwhenever   x  ≥  M. Suppose that   ϵ and M are fixed. Then   x  +  1  &amp;gt;  x  ≥  M, so by the definition of the statement       lim          x      →      ∞        f  (  x  )  =  L it must be the case that      |    f  (  x  +  1  )  −  L      |    &amp;lt;  ϵ  .. An identical proof holds for the limit of   f  (  x  +  2  )As an example, consider   f  (  x  )  =      1    x  . We have      lim          x      →      ∞            1    x    =      lim          x      →      ∞            1          x      +      1        =      lim          x      →      ∞            1          x      +      2        =  0.The idea here is that it doesn&#039;t matter if we shift the input x by some finite value, since we are allowing x to grow unboundedly large.

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New Questions in Calculus 1