The correct option is C π8 As x →∞ #160; , #160; cos (π8x) → 1 Looking at the rest of the part, x → #160;∞lim #160; π8x→ 0 x → #1 ...

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Substitution Method to Remove Indeterminate Form

limx→∞xcosπ/8...

Question

$lim x \to \infty x cos (\frac{π}{8 x}) sin (\frac{π}{8 x}) =$

π

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\frac{π}{2}

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\frac{π}{8}

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\frac{π}{4}

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Solution

The correct option is C $\frac{π}{8}$
As $x \to \infty, cos (\frac{π}{8 x}) \to 1$
Looking at the rest of the part,
$lim x \to \infty \frac{π}{8 x} \to 0$
$lim x \to \infty x . sin (\frac{π}{8 x})$
$= lim x \to \infty \frac{sin (\frac{π}{8 x})}{\frac{1}{x}}$
We multiply and divide by $\frac{π}{8}$
$= lim x \to \infty, \frac{sin (\frac{π}{8 x})}{\frac{π}{8 x}} \times \frac{π}{8 x}$
Now using the property of a limit,
$lim y \to 0, \frac{sin y}{y}$ $= 1$
We get, applying the limit,
$= 1 \times$ $\frac{π}{8}$
$=$ $\frac{π}{8}$

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