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Property 3

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

$L . H . L . = lim x \to 1^{-} (a [x + 1] + b [x - 1])$
$= lim h \to 0 (a [1 - h + 1] + b [1 - h - 1])$
$= a - b$
$R . H . L . = lim x \to 1^{+} (a [x + 1] + b [x - 1])$
$= lim h \to 0 (a [1 + h + 1] + b [1 + h - 1])$
$= 2 a$
$∵$ Limit exist
$∴ L . H . L . = R . H . L .$
$\Rightarrow a - b = 2 a$
$∴ a + b = 0$

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