The value of limx→1(tanxπ4)tanπx2 is

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Question

The value of

lim x \to 1 {(tan \frac{x π}{4})}^{tan \frac{π x}{2}}

is

Options:

A . e−2

B . e−1

C . e

D . 1

Answer: Option B
:
B

lim x \to 1 {(tan \frac{x π}{4})}^{tan \frac{π x}{2}} (1^{\infty} f r o m)

e^{lim x \to 1} (tan \frac{x π}{4} - 1) tan \frac{π x}{2}

= e^{lim x \to 1} (\frac{sin \frac{π x}{4} - cos \frac{π x}{4}}{cos \frac{π x}{4}}) \frac{2 sin \frac{π x}{4} cos \frac{π x}{4}}{cos \frac{π x}{2}}

= e^{lim x \to 1} \frac{2 {sin}^{2} \frac{π x}{4} - 2 sin \frac{π x}{4} cos \frac{π x}{4}}{cos \frac{π x}{2}}

= e^{lim x \to 1} (\frac{1 - cos \frac{π x}{2} - sin \frac{π x}{2}}{cos \frac{π x}{2}})

= e^{lim x \to 1} (\frac{1 - sin \frac{π x}{2}}{cos \frac{π x}{2}} - 1)

= e^{lim x \to 1} (\frac{cos \frac{π x}{2}}{1 + sin \frac{π x}{2}} - 1)

=

e^{- 1}

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