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What a difference two roots make ! (Posted on 2008-05-24) Difficulty: 2 of 5
Evaluate:

Limit ( (√(y + √(y + √y))) - √y)
y → ∞

See The Solution Submitted by K Sengupta    
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Solution re: Solution | Comment 3 of 5 |
(In reply to Solution by FrankM)

I must admit I don't follow FrankM's derivation. To me it just looks like 1/u * (1+u/2) - 1 is what is being said, but that's doesn't strike me as having a limit of 1/2.

The limit does however seem to be 1/2, as this repeated doubling of y shows:

            y     f(y)
              1 0.55377397403004
              2 0.54735699843337
              4 0.53958456106175
              8 0.53173024538280
             16 0.52461445374074
             32 0.51862290522359
             64 0.51382883162673
            128 0.51012856130407
            256 0.50734450184131
            512 0.50528737323993
           1024 0.50378675813856
           2048 0.50270201685996
           4096 0.50192293247069
           8192 0.50136592435133
          16384 0.50096897390958
          32768 0.50068673371316
          65536 0.50048637901274
         131072 0.50034431511696
         262144 0.50024366442744
         524288 0.50017239529913
        1048576 0.50012195118320
        2097152 0.50008625716939
        4194304 0.50006100536393
        8388608 0.50004314347525
       16777216 0.50003051012880
       33554432 0.50002157546159
       67108864 0.50001525692657
      134217728 0.50001078866195
      268435456 0.50000762892889
      536870912 0.50000539456379
     1073741824 0.50000381458085
     2147483648 0.50000269734010
     4294967296 0.50000190731953
     8589934592 0.50000134868461
    17179869184 0.50000095366704
    34359738368 0.50000067434594
    68719476736 0.50000047683534
   137438953472 0.50000033717387
   274877906944 0.50000023841812
   549755813888 0.50000016858718
  1099511627776 0.50000011920918
  2199023255552 0.50000008429367
  4398046511104 0.50000005960464
  8796093022208 0.50000004214689
 17592186044416 0.50000002980232
 35184372088832 0.50000002107345
 70368744177664 0.50000001490116
140737488355328 0.50000001053741

DEFDBL A-Z
y = 1
FOR i = 1 TO 48
 v = SQR(y + SQR(y + SQR(y))) - SQR(y)
 PRINT USING "############### #.##############"; y; v
 y = y * 2
NEXT


  Posted by Charlie on 2008-05-25 13:33:08
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