Question

Let $g\left(x\right)=\begin{cases} \frac{a x^{2} + b x + c \left( cot x \right)^{n}}{4 + \left( cot x \right)^{n}} & ,x\in \left(0 , \frac{\pi }{4}\right) \\ 1 & ,\text{ at }x=\frac{\pi }{4}, \\ \frac{sin x + cos x + \left( tan x \right)^{n}}{1 + c \left( tan x \right)^{n}} & ,x\in \left(\frac{\pi }{4} , \frac{\pi }{2}\right) , \end{cases}$ where $a,b,c$ are real constants and $f (x)=\displaystyle\lim _{n \rightarrow \infty} g (x) .$ If $\displaystyle\lim _{x \rightarrow \frac{\pi}{4}} f (x)$ exists, then $c$ may be equal to

• A. $2$
• B. $\frac{1}{2}$
• C. $3$
• D. $-1$

Answer 1

Answer: (D)

$g\left(x\right)=\begin{cases} \frac{\frac{a x^{2} + b x}{\left( cot x \right)^{n}} + c}{1 + \frac{4}{\left( cot x \right)^{n}}}=c\text{ if }x < \frac{\pi }{4} \\ \frac{\frac{sin x + cos x}{\left( tan x \right)^{n}} + 1}{\frac{1}{\left( tan x \right)^{n}} + c}=\frac{1}{c}\text{ if }x>\frac{\pi }{4} \end{cases}$
limit to exist $c=\frac{1}{c}\Rightarrow c^{2}=1,c=\pm1$