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  4. lim x arrow 3([x-3]+[3-x]-x), where [⋅] denotes the greatest integer function, is equal to

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Q. $\lim _{x \rightarrow 3}([x-3]+[3-x]-x)$, where $[\cdot]$ denotes the greatest integer function, is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 3}([x-3]+[3-x]-x)$
$=\displaystyle\lim _{x \rightarrow 3}([x]-3+3+[-x]-x)$
$=\displaystyle\lim _{x \rightarrow 3}([x]+[-x]-x)=-1-3=-4$
$(\because[x]+[-x]=-1$, when $x \notin I)$