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Substitution Method to Remove Indeterminate Form

limx → 1 [ co...

Question

$lim x \to 1 {[cosec \frac{π x}{2}]}^{\frac{1}{(1 - x)}} (where [.]$ represents the greatest integer function $)$ is equal to

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Solution

$lim x \to 1 {[cosec \frac{π x}{2}]}^{\frac{1}{(1 - x)}}$

$cosec \frac{π x}{2} \to 1$ when $x \to 1$
and
$[cosec \frac{π x}{2}] = 1$

$∴$ limit $= 1$

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