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If a,b∈ℝ and ...

Question

If $a, b \in R$ and $lim x \to 1 (a [x + 1] + b [x - 1])$ exists, then the value of $a + b$ is
$($ where $[.]$ denotes greatest integer function $)$

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Solution

$L . H . L . = lim x \to 1^{-} (a [x + 1] + b [x - 1]) = a - b R . H . L . = lim x \to 1^{+} (a [x + 1] + b [x - 1]) = 2 a$

$∵$ Limit exist
$L.H.L. = R.H.L. \Rightarrow a - b = 2 a ∴ a + b = 0$

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