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Getting Cubic With Limits (Posted on 2008-02-09) Difficulty: 2 of 5
Determine all possible real t that satisfy this relationship:

Lim (5p + 7p + 11p)1/p = t3 - t + 5
p→∞

  Submitted by K Sengupta    
Rating: 4.0000 (1 votes)
Solution: (Hide)
The only possible value of t is 2.

EXPLANATION:

(5p + 7p + 11p)1/p
= 11*((5/11)p + (7/11)p + 1)1/p)

Since each of (5/11)^p and (7/11)^p separately tends to zero whenever p tends to infinity, we must have:

Limit (5p +7p + 11p)1/p
p-> infinity

= 11*1 = 11

Hence, t^3 - t + 5 = 11
or, t^3 - t -6 = 0
or, (t-2)(t^2+t+3)=0

The only real root occurs at t=2, and so the required value of t is 2.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolutionDej Mar2008-02-09 21:42:01
SolutionHere it isFrankM2008-02-09 13:38:50
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