Question

Let $l=\underset{x\to 0}{lim}{\left(tan(\frac{\pi }{4}+x)\right)}^{\frac{1}{x}}$. Find $\left[l\right]$.($[.]$ denotes greatest integer function)

Answer 1

$l={lim}_{x\to 0}{\left(\tan (\frac{\pi }{4}+x)\right)}^{\frac{1}{x}}$

this of indeterminate form $1^{\infty }$

taking log on both sides

$\ln l={lim}_{x\to 0}\frac{1}{x}\ln \left(\tan (\frac{\pi }{4}+x)\right)$

Applying L'Hospitals rule:

$\ln l={lim}_{x\to 0}\frac{\frac{{\sec }^2(\frac{\pi }{4}+x)}{\tan \frac{\pi }{4}+x}}{1}$

$\ln l=\frac{{\sec }^2(\frac{\pi }{4}+0)}{\tan \frac{\pi }{4}+0}=\frac{{\sqrt{2}}^2}{1}=2$

Thus, $l=e^2$

i.e. $l={2.18}^2$

$\therefore l=7.3890$

Now, $\left[l\right]=\left[7.3890\right]=7$