limit of floor function, ceiling function and fractionsWe know that ⌊ x ⌋ : greatest integer ≤ x ⌈ x ⌉ : least integer ≥ x f r a c ( x ) = x − ⌊ x ⌋I can solve the questions limit of function like lim x → n ± ⌊ x − 1 ⌋ x − 1 lim x → n ± ⌊ x ⌋ x − 1 As x approaches n from above, ⌊ x − 1 ⌋ = n − 1; therefore, lim x → n + ⌊ x − 1 ⌋ x − 1 = n − 1 n − 1 but I can't solve the questions like lim x → ∞ ⌊ x − 3 ⌋ x − 1 , lim x → ∞ ⌈ x − 3 ⌉ x − 1 , lim x → a ⌊ x ⌋ , lim x → a ⌈ x ⌉ , lim x → a f r a c ( x )for a ∈ R All questions are similar type, so I have given many problems in my questions. Please help me.

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limit of floor function, ceiling function and fractionsWe know that  ⌊  x  ⌋ : greatest integer   ≤  x  ⌈  x  ⌉ : least integer   ≥  x      f    r    a    c    (  x  )  =  x  −  ⌊  x  ⌋I can solve the questions limit of function like      lim          x      →              n        ±                        ⌊      x      −      1      ⌋              x      −      1              lim          x      →              n        ±                        ⌊      x      ⌋              x      −      1      As   x approaches   n from above,   ⌊  x  −  1  ⌋  =  n  −  1; therefore,      lim                  x        →                  n          +                                ⌊      x      −      1      ⌋              x      −      1        =            n      −      1              n      −      1      but I can&#039;t solve the questions like      lim          x      →      ∞                  ⌊      x      −      3      ⌋              x      −      1        ,        lim          x      →      ∞                  ⌈      x      −      3      ⌉              x      −      1        ,        lim          x      →      a        ⌊  x  ⌋  ,        lim          x      →      a        ⌈  x  ⌉  ,        lim          x      →      a            f    r    a    c    (  x  )for   a  ∈      R  All questions are similar type, so I have given many problems in my questions. Please help me.

Ismael Blackwell · Accepted Answer

For the limit at infinity:  L  =      lim          x      →      ∞                  ⌊      x      −      3      ⌋              x      −      1        =      lim          x      →      ∞                  x      −      3      −      frac      (      x      −      3      )              x      −      1        =      lim          x      →      ∞                  x      −      3              x      −      1        −            frac      (      x      −      3      )              x      −      1      since  0  ≤            frac      (      x      −      3      )              x      −      1        &amp;lt;      1          x      −      1      we have  L  =      lim          x      →      ∞                  x      −      3              x      −      1        =  1For the limit at a:Let   ϵ  =            frac      (      a      )        2  , we have that   ⌊  x  ⌋  =  ⌊  a  ⌋ for   x  ∈  (  a  −  ϵ  ,  a  +  ϵ  )  . So       lim          x      →      a        ⌊  x  ⌋  =  ⌊  a  ⌋  .NOTE THIS ASSUMES   frac  (  a  )  &amp;gt;  0, if   a  ∈      Z   then the limit does not exist.

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