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  4. Evaluate displaystyle lim x arrow 2- x+(x-[x])2 where [ ] represents greatest integer function.

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Q. Evaluate $\displaystyle\lim _{x \rightarrow 2^{-}}\left\{x+(x-[x])^{2}\right\}$ where $[\,\,\, ]$ represents greatest integer function.

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 2^{-}}\left\{x+(x-[x])^{2}\right\}=\displaystyle\lim _{x \rightarrow 2^{-}}\left\{x+(x-1)^{2}\right\}$
$=\displaystyle\lim _{x \rightarrow 2^{-}}\left(x^{2}-x+1\right)$
$=4-2+1=3$