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

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

 See The Solution Submitted by K Sengupta No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 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 0.50000001490116140737488355328 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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