Prove that `lim_(xto2) [x]` does not exists, where [.] represents the greatest integer function.
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We know that greatest integer function changes its value at integral values of x. So, let us find LHL and RHL. `LHL=underset(xto2)lim[x]=underset(htoo)lim[2-h]` Sine h is positive and infinitely small `(2-h)` is value very close to 2 but smaller than 2 (say 1.99999) `:." "[2-h]=1,` when h approaches to zero. Thus, LHL=1 `RHL=underset(xto2)lim[x]=underset(hto0)lim[2+h]=[2.00001]=2` We observe that `LHLneRHL.` So, `underset(xto2)lim[x]` does not exist.
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