Let `f(x)=[x]=` Greatest integer less than or equal to x and k be an integer. Then, which one of the following in not correct?A. `lim_(xtok^-)f(x)=k-1`B. `lim _(xtok)f(x)=k`C. `lim _(xtok)f(x)"exists"`D. `lim_(xtok)f(x)` does not exist

Let `f(x)=[x]=` Greatest integer less than or equal to x and k be an i

1.	Let `f(x)=[x]=` Greatest integer less than or equal to x and k be an integer. Then, which one of the following in not correct?A. `lim_(xtok^-)f(x)=k-1`B. `lim _(xtok)f(x)=k`C. `lim _(xtok)f(x)"exists"`D. `lim_(xtok)f(x)` does not exist
Answer» Correct Answer - C We have, `f(x)=[x]` `therefore ("LHS at" x =k)` `=lim_(xtok^-)f(x)=lim_(hto0)=lim_(hto0)f(k-h)=lim_(hto0)[h-k]` `=lim_(hto0)k-1=k-1[because k-1ltk-hltk-hlt ktherefore [k-h]=k-1]` `therefore (RHL "at" x=k)` `=lim_(xto0)f(x)=lim_(hto0)f(k-h)=lim_(hto0)[k+h]` `=lim_(hto0)k=k[because klt k+hlt k+1therefore [k+h]=k]` Clearly, `lim_(xt0k^+)f(x) ne lim_(xtok^+)f(x). So, lim_(xtok)f(x)` does not exist.

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