Question

Let $g\left(x\right)=\{\begin{array}{ll}\frac{a{x}^2+bx+c{\left(cotx\right)}^n}{4+{\left(cotx\right)}^n}& ,x\in \left(0,\frac{\pi }{4}\right)\\ 1& ,\text{ at }x=\frac{\pi }{4},\\ \frac{sinx+cosx+{\left(tanx\right)}^n}{1+c{\left(tanx\right)}^n}& ,x\in \left(\frac{\pi }{4},\frac{\pi }{2}\right),\end{array}$ where $a,b,c$ are real constants and $f(x)=\underset{n\to \infty }{\lim }g(x).$ If $\underset{x\to \frac{\pi }{4}}{\lim }f(x)$ exists, then $c$ may be equal to

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

Answer 1

Answer: (D)

$g\left(x\right)=\{\begin{array}{l}\frac{\frac{a{x}^2+bx}{{\left(cotx\right)}^n}+c}{1+\frac{4}{{\left(cotx\right)}^n}}=c\text{ if }x<\frac{\pi }{4}\\ \frac{\frac{sinx+cosx}{{\left(tanx\right)}^n}+1}{\frac{1}{{\left(tanx\right)}^n}+c}=\frac{1}{c}\text{ if }x>\frac{\pi }{4}\end{array}$
limit to exist $c=\frac{1}{c}\Rightarrow c^2=1,c=\pm 1$