Question

The value of $\underset{x\to \pi /4}{lim}\frac{{\left(\tan x\right)}^{\tan x}-\tan x}{\ln \left(\tan x\right)-\tan x+1}$ is

• A. $-2$
• B. $1$
• C. $0$
• D. None of these

Answer 1

Answer: (A)

$-2$

We have,

$\underset{x\to \frac{\pi }{4}}{lim}\frac{{\left(\tan x\right)}^{\tan x}-\tan x}{\ln \left(\tan x\right)-\tan x+1}$

$=\underset{t\to 1}{lim}\frac{t^t-t}{\ln t-t+1}\left(\frac{0}{0}\right)$ $[$Substitute $\tan x=t]$

$=\underset{t\to 1}{lim}\frac{t^t(1+\ln t)-1}{\frac{1}{t}-1}\left(\frac{0}{0}\right)$

$=\underset{t\to 1}{lim}\frac{t^t{(1+\ln t)}^2+t^t\left(\frac{1}{t}\right)}{\frac{-1}{t^2}}=\frac{1+1}{-1}=-2$