The value of limx→1 tan⁡πx4tan⁡πx2 is

# The value of $\underset{\mathrm{x}\to 1}{lim} {\left(\mathrm{tan}\frac{\mathrm{\pi x}}{4}\right)}^{\mathrm{tan}\frac{\mathrm{\pi x}}{2}}$ is

1. A

e-2

2. B

e-1

3. C

e

4. D

1

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### Solution:

$\underset{\mathrm{x}\to 0}{lim} {\left(\mathrm{tan}\frac{\mathrm{\pi x}}{4}\right)}^{\mathrm{tan}\frac{\mathrm{\pi x}}{2}}$

$={\mathrm{e}}^{\underset{\mathrm{x}\to 1}{lim} \left(\mathrm{tan}\frac{\mathrm{\pi x}}{4}-1\right)\mathrm{tan}\frac{\mathrm{\pi x}}{2}}={\mathrm{e}}^{\underset{\mathrm{x}\to 1}{lim} \left(\frac{\mathrm{sin}\frac{\mathrm{\pi x}}{4}-\mathrm{cos}\frac{\mathrm{\pi x}}{4}}{\mathrm{cos}\frac{\mathrm{\pi x}}{4}}\right)\frac{2\mathrm{sin}\frac{\mathrm{\pi x}}{4}\mathrm{cos}\frac{\mathrm{\pi x}}{4}}{\mathrm{cos}\frac{\mathrm{\pi x}}{2}}}$

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