How Do You Approach This Limit Problem?

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SUMMARY

The discussion centers on the limit problem expressed as \(\lim_{n\to \infty}(\sqrt{n^{2}+n+1}-[n])\). Participants agree that for very large values of \(n\), the term \(n^2\) dominates the lower order terms, leading to the conclusion that the expression asymptotically approaches \(n - [n]\), which does not converge to a limit. The key takeaway is that the limit does not exist due to the behavior of the floor function as \(n\) approaches infinity.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with asymptotic analysis
  • Knowledge of the floor function notation \([n]\)
  • Basic algebraic manipulation of square roots
NEXT STEPS
  • Study the properties of limits involving square roots
  • Explore asymptotic notation and its applications
  • Learn about the behavior of the floor function in limits
  • Investigate advanced limit techniques, such as L'Hôpital's Rule
USEFUL FOR

Students of calculus, mathematicians, and anyone interested in advanced limit problems and asymptotic behavior in mathematical analysis.

aviravir1
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[tex]\lim_{n\to infinity}{{(\sqrt{n^{2}+n+1}})-[{(\sqrt{n^{2}+n+1}})]}[/tex] wer [.] is GIF
how will u solve this one
 
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For very large n, [itex]n^2[/itex] 'dominates' lower powers so that is asymptotic to [itex]n- [n][/itex]. And that has no limit.
 

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