Find the limit of fraction involving logarithmsI am looking for a way to prove the following limit for integer xs: lim x → ∞ log ⁡ ( x + 2 ) − log ⁡ ( x + 1 ) log ⁡ ( x + 2 ) − log ⁡ ( x ) = 1 2 I could find the result by using a computer program but I cannot formally establish the above equality.

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Find the limit of fraction involving logarithmsI am looking for a way to prove the following limit for integer   xs:      lim          x      →      ∞                          log        ⁡        (        x        +        2        )        −        log        ⁡        (        x        +        1        )                    log        ⁡        (        x        +        2        )        −        log        ⁡        (        x        )              =      1    2  I could find the result by using a computer program but I cannot formally establish the above equality.

Nia Molina · Accepted Answer

You may write, as   x  →  ∞            log      ⁡      (      x      +      2      )      −      log      ⁡      (      x      +      1      )              log      ⁡      (      x      +      2      )      −      log      ⁡      (      x      )        =            log      ⁡      (      1      +              2        x            )      −      log      ⁡      (      1      +              1        x            )              log      ⁡      (      1      +              2        x            )        =                    1        x            +      O      (      1              /                    x        2            )                      2        x            +      O      (      1              /                    x        2            )        =      1    2    +  O  (  1      /    x  )  .

Izabella Ponce · Accepted Answer

The Mean Value Theorem says,                                          log            ⁡            (            x            +            2            )            −            log            ⁡            (            x            +            1            )                                log            ⁡            (            x            +            2            )            −            log            ⁡            (            x            )                                              =                  1          2                                                    log              ⁡              (              x              +              2              )              −              log              ⁡              (              x              +              1              )                        1                                              log              ⁡              (              x              +              2              )              −              log              ⁡              (              x              )                        2                                                            =                  1          2                                      1                          ξ              1                                            1                          ξ              0                                                                        =                  1          2                                      ξ            0                                ξ            1                              where   x  +  1  &amp;lt;      ξ    1    &amp;lt;  x  +  2 and   x  &amp;lt;      ξ    0    &amp;lt;  x  +  2. Therefore,      1    2        x          x      +      2        ≤            log      ⁡      (      x      +      2      )      −      log      ⁡      (      x      +      1      )              log      ⁡      (      x      +      2      )      −      log      ⁡      (      x      )        ≤      1    2              x      +      2              x      +      1      The Squeeze Theorem, then says that      lim          x      →      ∞                  log      ⁡      (      x      +      2      )      −      log      ⁡      (      x      +      1      )              log      ⁡      (      x      +      2      )      −      log      ⁡      (      x      )        =      1    2

Find the limit of fraction involving logarithms I am looking for a way to prove the following limit

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