One Sided Limits of Greatest Integer Function

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SUMMARY

The discussion focuses on finding one-sided limits for the greatest integer function, specifically Lim x->n-, [[x/2]]. When n is even, the limit approaches n/2, while for odd n, the limit approaches (n-1)/2. The continuity of the greatest integer function between integer points allows for the application of epsilon-delta proofs to establish these limits rigorously. Participants emphasize the importance of understanding the properties of the greatest integer function in limit calculations.

PREREQUISITES
  • Understanding of the greatest integer function (floor function)
  • Familiarity with one-sided limits in calculus
  • Knowledge of epsilon-delta definitions of limits
  • Basic concepts of continuity in mathematical functions
NEXT STEPS
  • Study the properties of the greatest integer function in detail
  • Learn about epsilon-delta proofs for limit calculations
  • Explore continuity and discontinuity in piecewise functions
  • Practice finding one-sided limits for various functions
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Students and educators in calculus, mathematicians interested in limit theory, and anyone studying piecewise functions and their properties.

aliasadullahb
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Can somebody tell me how to find one sided limits for greatest integer function, say Lim x->n-, [[x/2]]. that is limit x approaches n from left of [[x/2]] where [[]] represent greatest integer function and n is any integer.

I know how to find one sided limits for simple [[x]].
 
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If n is even, the limit is n/2. If n is odd, the limit is (n-1)/2.
 
The technical way is to use the epsilon-delta proof to show that you get all the properties for some delta and epsilon and so on.

Since the function is continuous in-between the values of the integer points for these types of functions (floor, ceil, etc) then you can use math-man's answers provided above.
 

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