Question

Using L' Hospital's rule, evaluate:
${lim}_{x\to \frac{\pi }{4}}(xtanx-\frac{\pi }{4}secx)$

Answer 1

We have,

$f\left(x\right)={lim}_{x\to \frac{\pi }{4}}(x\tan x-\frac{\pi }{4}\sec x)$

Using not L; hospital rule because its not make indeterminate forms.

Then,

$f\left(x\right)={lim}_{x\to \frac{\pi }{4}}(x\tan x-\frac{\pi }{4}\sec x)$

$f\left(x\right)=(\frac{\pi }{4}\tan \frac{\pi }{4}-\frac{\pi }{4}\sec \frac{\pi }{4})$

$=(\frac{\pi }{4}\times 1-\frac{\pi }{4}\times \sqrt{2})$

$=\frac{\pi }{4}(1-\sqrt{2})$

Hence, this is the answer.