Question

Find the left hand limit and right hand limit of the greatest integer function f(x) = [x] = greatest integer less than or equal to x at x = k, where k is an integer. Also, show that \(\lim\limits_{x \to k}\) f(x) does not exit.

Answer 1

We have−

f(x) = [x]

L.H.L (of x = k)

\(=\lim\limits_{x \to k-}f(x)\)

\(=\lim\limits_{h \to 0}f(k-h)\) 

\(=\lim\limits_{h \to 0}[k-h]\)

\(=\lim\limits_{h \to 0}(k-1)\)   [∵ k − 1 < k − h < k

⇒ [k − h] = k − 1]

= k-1

R.H.L (of x = k)

= \(\lim\limits_{h\to 0}f(k+h)\)

\(=\lim\limits_{h\to 0}(k+h)\) 

\(=\lim\limits_{h\to 0}k\)   [∵ k < k + h < k + 1

⇒ [k + h] = k]

= k

Here, L.H.L ≠ R.H.L

∴ \(\lim\limits_{x \to k}f(x)\) does not exit.