InterviewSolution
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InterviewSolution
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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. |
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Answer» 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. |
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