f(x) = [x], the greatest integer less then or equal to x and k is an integer. Then, ${lim}_{x \to k} f (x) =$ ___

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k - 1

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k + 1

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does not exist

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Solution

The correct option is D
does not exist

L.H.L = ${lim}_{x \to k^{-}} f (x) = {lim}_{h \to 0} f (k - h) = {lim}_{h \to 0} [k - h] = k - 1$
R.H.L = ${lim}_{x \to k^{+}} f (x) = {lim}_{h \to 0} f (k + h) = {lim}_{h \to 0} [k + h] = k$
L.H.L $\neq$ R.H.L. The limit does not exist for integer values of k.

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